LOANED   TO 
UNIVERSITY   OF    CALIFORNIA 

DEPARTMENT    OF     MECHANICAL     AND     ELECTRICAL     ENGINEERING 
FROM     PRIVATE    LIBRARY    OF 

C.    L.    CORY 


ALTERNATING  CURRENT 
MOTORS.      :-  -•'•:': 


BY 


A.  s.  MCALLISTER,  PH.D 


NEW  YORK 

McGRAW  PUBLISHING  COMPANY 
1906 


Engineering 
Library 


Copyrighted  1906 

by  the 

McGRAw  PUBLISHING  COMPANY 
New  York 


T/<z7<?/ 
A/3 


Library 


PREFACE. 

In  the  preparation  of  the  explanatory  matter  contained  in 
the  present  volume,  the  writer  has  had  constantly  in  mind  the 
difficulties  which  he  encountered  in  endeavoring  to  ascertain 
the  causes  for  the  various  effects  which  were  described  in  the 
available  electrical  literature  in  connection  with  the  operation 
of  the  several  types  of  alternating-current  motors.  An  exper- 
ience of  several  years  in  instructing  engineering  students  in 
alternating-current  phenomena  has  served  to  convince  the 
writer  that  the  above  mentioned  difficulties  are  due  primarily 
to  the  methods  employed  in  presenting  the  facts  to  minds  un- 
familiar therewith.  It  is  believed  that  a  large  part  of  the 
difficulty  may  be  attributed  to  an  inconsistent  use  of  technical 
terms,  such  as  the  word  "  field,"  meaning  at  one  time 
magnetism,  at  another  iron,  and  yet  at  another  copper. 
Many  writers  use  the  term  "  power  "  without  discrimination  for 
quantities  measurable  in  watts  or  in  watt -hours.  It  is  surpris- 
ing how  frequently  even  well-informed  writers  use  the  term 
"  current  "  when  "  e.m.f."  is  intended.  One  often  sees  the  state- 
ment "  a  current  is  induced,"  when,  as  a  matter  of  fact,  the 
circuit  in  which  the  current  is  supposed  to  flow  is  entirely  open. 
Much  of  the  difficulty  is  due  to  the  excessive  use  of  mathematical 
questions  as  an  end  rather  than  a  means.  It  is  believed  that 
mathematics  should  be  employed  merely  as  a  short-hand  method 
of  stating  facts.  Beyond  any  doubt  much  confusion  is  created 
in  the  mind  of  an  engineering  student  when  an  attempt  is  made 
to  express  certain  assumed  relations  in  the  form  of  mathematical 
equations,  and  then  to  transform  the  equations  to  obtain  a 
result  which  he  is  asked  to  believe  expresses  physical  facts, 
merely  because  the  equations,  considered  mathematically,  have 
been  properly  transformed. 

In  view  of  the  facts  stated  above,  and  on  account  of  the 
belief  that,  for  the  purpose  of  imparting  information,  simplicity 
is  the  criterion  of  worth,  an  attempt  has  been  made  to  deal 
directly  with  the  electromagnetic  phenomena  of  alternating- 
current  motors  in  the  simplest  possible  manner,  in  so  far  as  such 
method  is  consistent  with  accuracy.  Wherever  mathematical 


789531 


iv  PREFACE. 

equations  are  employed,  the  assumption  upon  which  they  are 
based  are  definitely  stated,  and  the  inaccuracies  in  the  assump- 
tions and  limitations  in  the  equations  are  carefully  noted,  while 
the  results  obtained  from  the  transformed  equations  are  inter- 
preted with  due  regard  to  the  inaccuracies  and  the  limitations. 
Moreover,  the  significance  of  each  transformation  is  carefully 
explained,  so  that  the  reader  may  be  constantly  reminded  of 
the  fact  that  he  is  dealing  directly  with  electromagnetic  phe- 
nomena and  only  indirectly  with  mathematics. 

It  has  been  assumed  that  the  reader  is  familiar  with  the  funda- 
mental facts  of  electricity  and  magnetism,  and  that  he  has  some 
knowledge  of  the  lower  branches  of  mathematics.  No  attempt 
has  been  made  to  explain  the  manner  in  which  alternating 
electromotive  forces  and  currents  may  be  represented  in  value 
and  time-phase  position  by  straight  lines,  because  a  knowledge 
of  such  representation  can  be  presupposed.  Much  attention  is 
given  to  graphical  diagrams  and  to  examination  of  the  facts  upon 
which  they  are  based.  Where  necessary  the  errors  involved  in  the 
assumptions  are  pointed  out,  and  the  magnitude  of  their  effect  on 
the  final  results  are  discussed.  In  developing  the  graphical  dia- 
gram of  the  induction  motor,  the  proof  that  the  current  locus  is 
(approximately)  a  circle  has  been  based  on  the  relation  between 
the  equivalent  electric  circuits  considered  as  certain  intercon- 
nected resistances  and  reactances,  rather  than  on  the  resolution 
of  the  magnetic  flux  into  certain  components.  The  latter 
method  is  the  one  most  commonly  employed.  It  is  believed, 
however,  that  the  former  method  possesses  peculiar  advantages 
in  permitting  the  reader  to  follow  the  development  step  by  step 
without  losing  sight  of  the  involved  electromagnetic  relations, 
and  in  allowing  the  inaccuracies  in  the  assumptions  to  be  defi- 
nitely determined. 

The  major  portion  of  the  volume  has  appeared  as  articles  in 
various  publications,  notably  the  Electrical  World,  the  American 
Electrician  and  the  Sibley  Journal  of  Engineering.  The  writer's 
thanks  are  due  to  the  editors  of  these  publications  for  permis- 
sion to  incorporate  the  articles  in  this  book.  The  writer 
wishes  to  take  this  opportunity  to  express  his  appreciation  of 
the  encouragement  continuously  received  from  his  friend  and 
former  colleague,  Dr.  Frederick  Bedell. 


CONTENTS. 


CHAPTER  I. 

SINGLE-PHASE  AND  POLYPHASE  CIRCUITS 1 

Economy  of  Conducting  Material 1 

Three-Phase  Power  Measurements 3 

Measuring  Three-Phase  Power  with  One  Wattmeter 6 

Wattmeters  on  Unbalanced  Loads 9 

Equivalent  Single-Phase  Currents 13 

Equivalent  Single-Phase  Resistance 14 

CHAPTER  II. 

OUTLINE  OF  INDUCTION  MOTOR  PHENOMENA 16 

Methods  of  Treatment 16 

Production  of  Revolving  Field 17 

Simple  Analytical  Equations 19 

Starting  Devices  for  Induction  Motor 22 

Concatenation  Control 23 


CHAPTER  III. 

OBSERVED  PERFORMANCE  OF  INDUCTION  MOTOR 25 

Test  with  One  Voltmeter  and  One  Wattmeter 25 

Measurement  of  Slip 26 

Determination  of  Torque.  . 27 

Measurement  of  Secondary  Resistance 28 

Determination  of  Secondary  Current 28 

Calculation  of  Primary  Power  Factor 29 

Calculation  of  Primary  Current 30 

CHAPTER  IV. 

INDUCTION  MOTORS  AS  FREQUENCY  CONVERTERS 31 

Field  of  Application 31 

Characteristic  Performance 32 

Capacity  of  Frequency  Converters 33 

v 


vi  CONTENTS. 

CHAPTER  V. 

THE  HEYLAND  INDUCTION  MOTOR 35 

Excitation  of  Induction  Motors 35 

Secondary  Exciting  Magnetomotive  Force 37 

Direct  Current  in  Secondary  Coils 38 

Alternating  Current  in  Secondary  Coils 39 

Action  with  Stationary  Rotor 41 

Action  with  Rotor  at  Synchronous  Speed 43 

Function  of  the  Connecting  Resistances 44 

Double  Three-Phase  Secondary  Winding 46 

CHAPTER  VI. 

THE  SINGLE-PHASE  INDUCTION  MOTOR 48 

Outline  of  Characteristic  Features 48 

Production  of  Quadrature  Magnetism 49 

Production  of  Revolving  Field 51 

Elliptical  Revolving  Field 52 

Starting  Torque  of  the  Single-Phase  Motor 53 

Use  of  "  Shading  Coils  " 54 

Use  of  Commutator  on  the  Rotor 56 

Polyphase  Induction  Motors  Used  as  Single-Phase  Machines. ...  58 

CHAPTER  VII. 

GRAPHICAL  TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA 63 

Advantage  of  Graphical  Methods -. 63 

Effect  of  Inserting  Resistance  in  the  Secondary 63 

Primary  and  Secondary  Current  Locus 65 

Test  Results 66 

Equation  of  the  Current  Locus 70 

Errors  in  Assuming  a  Circular  Arc 71 

Effect  of  Design  on  Leakage  Reactance 73 

CHAPTER  VIII. 

INDUCTION  MOTORS  AS  ASYNCHRONOUS  GENERATORS 74 

Operation  Below  Synchronism 74 

Operation  Above  Synchron  sm 75 

Vector  Diagram  of  Currents 75 

Performance  Characteristics 79 

Parallel  Operation  of  Asynchronous  Generators 80 

Excitation  of  Asynchronous  Generators 81 

Condensance  as  a  Source  of  Exciting  Current 83 

Condensers  in  Alternating  Current  Circuits 84 

Excitation  Characteristics  of  Asynchronous  Generators 86 

Load  Characteristics  of  Asynchronous  Generators 90 

Commutator  Excitation  of  Asynchronous  Generators 92 


CONTENTS.  vii 

CHAPTER  IX. 

TRANSFORMER  FEATURES  OP  THE  INDUCTION  MOTOR 95 

Electric  and  Magnetic  Circuits .  .  95 

Equivalent  Electric  Circuits 98 

Modified  Electric  Circuits 99 

Circle  Diagram  of  Currents 100 

Internal  Voltage  Diagrams  of  the  Induction  Motor 105 

Corrected  Current  Locus  of  the  Induction  Motor 107 

Complete   Performance   Diagram   of   the   Polyphase   Induction 

Motor 109 

Comparison  of  Single-Phase  and  Polyphase  Motors 112 

Electric  Circuits  of  the  Single-Phase  Induction  Motor 114 

Complete  Performance  Diagram  of  the  Single-Phase  Induction 

Motor 1 15 

Speed  and  Torque  of  the  Single-Phase  Induction  Motor. ......  117 

Capacities  of  Single-Phase  and  Polyphase  Motors 119 

CHAPTER  X. 

MAGNETIC  FIELD  IN  INDUCTION  MOTORS 121 

Polyphase  Motors 121 

Magnetic  Distribution  with  Open  Secondary 124 

Magnetic  Distribution  with  Closed  Secondary 127 

Determination  of  Core  Flux 130 

Effect  on  Core  Flux  of  Using  Distributed  Winding 131 

Effect  on  Capacity  of  Varying  the  Grouping  of  Coils 133 

Exciting  Watts  in  Induction  Motors 135 

Magnetic  Field  in  the  Single-Phase  Induction  Motor 138 

Production  of  Speed-Field  Current 139 

Transformer  Features  of  the  Single-Phase  Induction  Motor.  .  .  .  142 

Secondary  Currents  in  the  Single-Phase  Motor 143 

Graphical  Representation  of  Secondary  Quantities 146 

CHAPTER  XI. 

SYNCHRONOUS  MOTORS  AND  CONVERTERS 149 

Synchronous  Commutating  Machines 149 

Synchronous  Motors  and  Generators 151 

Synchronous  Converters,  Unity  Power-Factor 154 

Distribution  of  Heat  Loss  in  Armature  Coils 159 

Synchronous  Converters,  Fractional  Power- Factor 160 

Double  Current  Machines 161 

Relative  Capacities  of  Synchronous  Machines  of  Various  Phases.  163 

Characteristic  Performance  of  Synchronous  Converters 165 

Excitation  of  Synchronous  Machines 166 

Hunting  of  Synchronous  Machines 167 

Starting  of  Synchronous  Converters 168 


viii  CONTENTS. 

Compounding  of  Synchronous  Converters 169 

Inverted  Converters 170 

Predetermination  of  Performance  of  Synchronous  Converters.  .  171 

SixrPhase  Converters 174 

Six-Phase  Transformation 175 

Relative  Advantages  of  Delta  and  Star-Connected  Primaries .  .  180 


CHAPTER  XII. 

ELECTROMAGNETIC  TORQUE 181 

Commutator  Motors 181 

Equality  of  Torques  for  Uniform  Reluctance 181 

Inequality  of  Torques  for  Non-Uniform  Reluctance 183 

Determination  of  Torque  by  Calculation  of  the  Output 184 

Measurement  of  Torque  by  the  Loading-Back  Method 185 

Elimination  of  Errors ...  187 


CHAPTER  XIII. 

SIMPLIFIED  TREATMENT  OF  SINGLE-PHASE  COMMUTATOR  MOTORS.  .  .  189 

The  Repulsion  Motor  ..........................  ;y  ..........  184 

Electric  and  Magnetic  Circuits  of  Ideal  Motor  .....  *....>.  ......  190 

Production  of  Rotor  Torque  ......................  '^$  ......  192 

Graphical  Diagram  of  Repulsion  Motor  ............  .-...•  ......  193 

Calculated  Performance  ot  Ideal  Repulsion  Motor  .  .  .  .........  196 


CHAPTER  XIV. 

MOTORS  OF  THE  REPULSION  TYPE  TREATED  BOTH  GRAPHICALLY  AND 

ALGEBRAICALLY  ...................  .  .........................  199 

Electromotive  Forces  Produced  in  an  Alternating  Field  .......  199 

The  Simple  Repulsion  Motor  ..............................  201 

Effect  of  Speed  on  the  Stator  Electromotive  Forces  ...........  203 

Fundamental  Equations  of  the  Repulsion  Motor  ..............  204 

Vector  Diagram  of  Ideal  Repulsion  Motor  ...................  206 

Corrections  for  Resistance  and  Local  Leakage  Reactance  ......  209 

Brush  Short-Circuiting  Effect  .............................  .  212 

Observed  Performance  of  Repulsion  Motor  ...................  213 

Compensated  Repulsion  Motor  .............................  214      V 

Apparent  Impedance  of  Motor  Circuits.  .  ....................  217     '** 

Fundamental  Equations  of  the  Compensated  Repulsion  Motor.  .  218 

Vector  Diagram  of  Compensated  Repulsion  Motor  ............  221 

Calculated  Performance  of  Compensated  Repulsion  Motor  .....  223 

Observed  Performance  of  Compensated  Repulsion  Motor  ......  225 

Corrections  for  Resistance  and  Local  Leakage  Reactance  ......  227 

Brush  Short-Circuiting  Effect  ..............................  228 


CONTENTS.  ix 

CHAPTER  XV. 

MOTORS  OF  THE  SERIES  TYPE  TREATED  BOTH  GRAPHICALLY  AND  AL- 
GEBRAICALLY   232 

The  Plain  Series  Motor 232 

Fundamental  Equations  of  Series  Motor  with  Uniform  Air-Gap 

Reluctance 233 

Fundamental  Equations  for  Motor  with  Non-Uniform  Reluctance  237 

Inductively  Compensated  Series  Motor 241 

Conductively  Compensated  Series  Motor 242 

Complete  Performance  Equations  of  Compensated  Motors 243 

Vector  Diagram  of  Compensated  Series  Motor 244 

Induction  Series  Motor 245 

Fundamental  Equations  of  Induction  Series  Motors 247 

Corrections  for  Resistance  and  Local  Leakage  Reactance 252 

Vector  Diagram  of  Induction  Series  Motor 253 

Generator  Action  of  Induction  Series  Motor 254 

Brush  Short-Circuiting  Effect 255 

Hysteretic  Angle  of  Time-Phase  Displacement 257 

Power  Factor  of  Commutator  Motors 260 

Resistance  in  Shunt  with  Field  Winding 260 

Loss  Due  to  Use  of  Shunted  Resistance. .                                       .  263 


CHAPTER  XVI. 

PREVENTION  OF  SPARKING  IN  SINGLE-PHASE  COMMUTATOR  MOTORS.  .  266 

Transformer  Action  with  Stationary  Rotor 266 

Interlaced  Armature  Windings 267 

Use  of  Series  Resistance 267 

Power  Lost  in  Resistance  Leads 268 

Internal  Resistance  Leads 269 

External  Resistance  Leads  with  Two  Commutators 269 

External  Resistance  Leads  wich  One  Commutator 270 


CHAPTER  I. 
SINGLE-PHASE  AND   POLYPHASE   CIRCUITS. 

ECONOMY  OF  CONDUCTING  MATERIAL. 

Although  of  all  possible  transmission  systems  the  single-phase 
requires  the  least  number  of  conductors  and,  on  the  basis  of 
equality  of  maximum  e.m.f  .  to  the  neutral  point,  the  single- 
phase  is  the  equal  with  reference  to  the  cost  of  conductors  of 
any  polyphase  system,  practically  all  of  the  long-distance  trans- 
mission circuits  are  of  the  polyphase  type,  chiefly  because  of  the 
facts  that  polyphase  generators  are  less  expensive  in  construc- 
tion and  more  economical  in  operation  than  single-phase  ma- 
chines, and,  for  the  purpose  of  power  distribution,  polyphase 
motors  and  synchronous  converters  are  superior  to  single-phase 
machines.  Of  all  polyphase  circuits  operated  with  a  given 
maximum  measurable  e.m.f.  between  lines,  the  three-phase  sys- 
tem is  the  most  economical  with  reference  to  the  cost  of  con- 
ducting material;  which  accounts  for  the  fact  that,  with  very 
few  exceptions,  all  transmission  circuits  are  of  the  three-phase 
type. 

That  all  symmetrical  transmission  systems  show  the  same 
economy  of  conducting  material,  irrespective  of  the  number  of 
phases,  when  compared  on  the  basis  of  equality  of  maximum 
e.m.f.  to  the  neutral  point,  can  be  proved  as  follows:  Let  P  = 
the  number  of  phases  (and  conducting  wires)  ;  R  =  the  resist- 
ance of  the  total  mass  of  conducting  material,  all  wires  being 
considered  in  parallel;  E  =  the  e.m.f.  to  the  neutral  point, 
and  W  =  the  power  transmitted.  Then  PR  =  resistance  per 

W  W 

wire;    —  ==  power  per  phase,   and  =   current   per   wire. 

Representing  the  current  per  wire  by  7,  the  loss  in  transmission 


which  is  obviously  independent  of  the  number  of  phases. 

1 


ALTERNATING  CURRENT  MOTORS. 


*•»"  t  OP  -aH'  systems  in  which  the  measurable  maximum  e.m.f.  is 
twice  the  e.m.f.  from  a  single  conductor  to  the  neutral  point, 
the  cost  of  conducting  material  for  transmitting  at  a  given 
loss  is  the  same.  In  this  class  fall  the  two-wire  (single-phase), 
four-wire  (two-phase),  six-wire  (six-phase),  and  other  systems  in 
which  the  number  of  phases  is  an  even  one.  An  inspection  of 
Fig.  1  will  show  that  in  the  three-phase  system  the  measurable 
e.m.f.  is  only 


or  1.732  times  that  from  one  conductor  to  the  neutral  point. 
On  the  basis  of  equality  of  measurable  e.m.f.,  therefore,  the 


three-phase    system    requires 


f  as  much  conducting 


FIG.  1.— E.m.f.'s  of      FIG.  2.— E.m.f.'s  of 
Three-phase  Circuits.      Six-phase  Circuits. 

material  as  any  system  in  which  the  number  of  phases,  P,  is 
even,  since  the  conducting  material  varies  inversely  as  the 
square  of  the  e.m.f.  for  any  given  system.  The  calculations  re- 
corded in  Table  I  show  that  a  higher  conductor  efficacy  is 
obtained  in  each  case  when  P  is  odd  than  when  it  is  even,  and 
that  the  smaller  the  value  of  P  the  higher  the  efficacy  will  be. 
Since  P,  when  odd,  cannot  be  less  than  three,  the  three-phase 
is  of  all  systems  the  most  economical. 

While  as  regards  the  desirability  of  obtaining  economy  in 
cost  of  conducting  material,  the  problem  of  determining  the 
proper  circuits  for  the  distribution  of  electrical  energy  at  the 
end  of  a  transmission  line  is  quite  similar  to  that  connected 
with  the  transmission  circuits,  factors  other  than  economy 
enter  into  this  portion  of  the  problem,  which  must  be  carefully 
considered  before  any  attempt  is  made  at  a  definite  solution. 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  3 

TABLE  I. — Relative  conducting  material  required  for  different  tranmisssion  systems 


8 

On  basis  of 

On  basis  of  max- 

«J 
JH 

minimum  e.m.f. 

imum  measurable  e.m.f. 

M 

°| 

11 

*S'C 

.0 

•§! 

ii 

a 

1 

P 

1             1 

^S 

ijS, 

p 

Zls 

H 

^ 

« 

1 

5^ 
S  <^ 

8 

|u 

<N 

c^                            w 

SL 

2 

2.0000                      1 

.000 

2.0000 

1.000 

3 

1.7320 

.750 

1.7320 

.750 

4 

1.4142 

.500 

2.0000                   

1.000 

5 

1.1756 

.346 

1.9022 

.905 

6 

1.0000 

.250 

2.0000 

1.000 

7 

.8678 

.188 

1.9500 

.950 

8 

.7654 

.146 

2.0000 

1.000 

9 

.6840 

.117 

1  .9696 

.969 

THREE-PHASE  POWER  MEASUREMENTS. 

In  measuring  the  power  in  a  three-phase  circuit  the  most 
convenient  method  is  one  involving  the  use  of  two  wattmeters 
whose  current  coils  are  placed  in  any  two  of  the  three-phase 
leads,  and  whose  pressure  coils  are  connected,  respectively, 
between  these  two  leads  and  the  third  lead.  A  simple  semi- 
graphical  proof  of  the  correctness  of  the  two-wattmeter  method 
of  measuring  the  power  in  a  three-phase  circuit  under  any 
condition  of  service  is  given  below. 

In  Fig.  3,  let  the  sides  of  the  triangle  ABC  represent  the 
relative  values  and  phase  positions  of  the  three  e.m.fs.  of  an 
unsymmetrical  three-phase  system.  Assume  the  receiver  to  be 
delta  connected,  and  let  IAB  be  the  current  in  coil  AB,  and  OAB 
be  the  angle  of  lag  of  this  current  with  respect  to  the  e.m.f. 
of  the  coil.  Similarly,  let  IBC  and  I  AC  represent  the  value  and 
phase  position  of  the  currents  in  coils  B  C  and  A  C,  No  restric- 
tion is  made  as  to  the  values  of  currents,  e.m.fs.  or  as  to  the 
several  lag  angles. 


4  ALTERNATING  CURRENT  MOTORS. 

Let  a  wattmeter  be  connected  with  its  current  coil  in  lead  at 
A,  and  its  e.m.f.  coil  across  between  this  lead  and  lead  C.  Also  a 
wattmeter  at  B,  with  pressure  coil  between  B  and  C.  Each 


FIG.  3. — Phase  Relation  of  e.m.f. 's  and  currents. 

wattmeter  will  show  a  deflection,  which  may  be  represented  by 
W  =  E  I  Cos  6,  where  /  is  the  amperes  in  the  current  coil,  E  is 
the  volts  across  the  e.m.f.  coil,  and  0  is  the  angle  between  this 
e.m.f.  and  the  current  7. 


FIG.  4. — Measurement  of  power  in  coils  A  C  and  C  B. 

For  sake  of  simplicity  in  explanation,  assume,  in  the  first 
place,  the  current  flowing  in  coil  A  B  to  be  absent  while  measure- 
ments are  made  upon  the  watts  supplied  to  the  other  coils,  as 
indicated  by  Fig.  4.  Evidently  the  sum  of  the  readings  of  the 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  5 

meters  as  connected  gives  the  watts  in  the  two  remaining  coils, 
since  each  meter  is  connected  as  though  measuring  power  in  a 
single-phase  circuit.  Now  assume,  in  the  second  place,  the 
current  to  flow  in  coil  A  B  alone  while  the  currents  in  the  other 
two  coils  are  absent,  as  shown  by  Fig.  5.  According  to  proof 
given  below,  the  wattmeters  as  connected  now  register  as  their 
sum  the  true  watts  supplied  to  coil  A  B.  When  all  three  cur- 
rents flow  simultaneously,  each  wattmeter  will  show  a  de- 
flection equal  to  the  sum  of  its  two  previous  readings,  since  its 
e.m.f.  coil  has  undergone  no  change  in  connection  and  the  two 
currents  causing  the  former  deflections  are  now  superposed,  and 
the  true  power  transferred  will  be  properly  recorded  by  the 
two  meters. 


FIG.  5. — Measuring  power  in  coil  A  B. 


Two  wattmeters  having  their  current  coils  in  series  with  a 
given  single-phase  load,  and  one  terminal  of  the  e.m.f.  coil  of 
each  meter  connected  to  the  opposite  leads  of  the  circuit  supply- 
ing power  to  the  load  and  the  other  two  free  terminals  con- 
nected together  and  placed  at  any  point  of  any  relative  poten- 
tial compared  with  that  of  the  load,  as  depicted  in  Fig.  5,  will 
give  the  true  value  of  power  transmitted. 

In  Fig.  6,  let  EAB  be  the  e.m.f.  across  the  load,  TAB  be  the 
load  current  and  QAB  be  the  angle  between  I  A  B  and  EAB.  Evi- 
dently the  watts  transmitted  are 

WAB  =  EAB  IAB  cos  OAB* 


6          ALTERNATING  CURRENT  MOTORS. 

Now  assume  a  wattmeter  connected  at  A  to  C.     Its  reading 
will  be 

WAC  =  SAC  IAB  cos  6AC. 
A  wattmeter  at  B  to  C  will  read 

WBC  =  EEC  IAB  cos  OBC- 
From  Fig.  6, 

EABcosdAB  =  AF. 
EAC  cos  QAC  =  A  E. 
EEC  cos  OBC  =  B  D  =  E  F. 
and  since  A  F  =  A  E  +  EFt 

IAB  (EAC  cos  6  AC  +  EEC  cos  BBC)  =  IAB  EAB  cos  OAB 
or  WAC  +  WBC  =  WAB 

and  this  value  is  independent  of  the  position  of  the  point  C. 


FIG.  6. — Graphical  proof  of  Fig.  5. 

In  consequence  of  this  fact,  P  —  1  wattmeters  may  be  used  to 
determine  the  true  power  in  any  P — phase  system,  however  un- 
symmetrical  may  be  the  phase  relations,  provided  the  free  ter- 
minals of  the  e.m.f.  coil  of  each  meter  be  connected  to  that 
lead  in  which  no  wattmeter  is  placed. 

MEASURING  THREE-PHASE  POWER  WITH  ONE  WATTMETER. 

A  single  wattmeter  method  which  may  be  used  for  deter- 
mining the  angle  of  lag  in  balanced  three-phase  circuits  is  out- 
lined below.  It  is  believed  that  this  method  is  not  as  generally 
well  known  as  its  simplicity  and  comparative  freedom  from 
errors  justify. 

If  a  wattmeter,  connected  as  indicated  by  the  diagram  of 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  7 

Fig.  7,  with  its  current  coil  in  one  lead  of  a  three-phase  circuit, 
be  read  with  its  e.m.f.  coil  across  between  this  lead  and  first  one 
and  then  the  other  of  the  remaining  two  leads,  the  two  values 
thus  obtained  may  be  used  to  determine  the  angle  of  lag  of  the 
current  by  means  of  the  following  relation: 

W1-W2 


where  0  is  the  angle  of  lag,  and  Wlt  W2  are  two  readings  of  the 
wattmeter,  as  indicated  above. 

When  <£  is  greater  than  60  degs.  one  reading  will  be  negative 
so  that  the  difference  of  readings  will  be  greater  than  their  sum. 


FIG.  7.— Measuring  three-phase  power  with  one  wattmeter. 

The  proof  of  this  relation  is  as  follows:   I 

Let  I  =  current  in  lead  A  X, 

E=  e.m.f.  of  A  B  and  of  A  C, 


then 


and  since 


and 


hence 


Wl  =  I E  cos  (0  -  30) 
W2  =  I  E  cos  (0  +  30) 

cos  (a  ±  ft)  =  cos  a  cos  ft  +  sin  a  sin  ft 

Wl-W2  =  IE  [cos  ( 0  -30)  -  cos  (0  +  30)  ] 

—  2  I  E  sin  30  sin  0  =  /  E  sin  0 

V,  +  W 2  =  /  E  [cos  (0  -  30°)  +  cos  (0  +  30)  ] 
=  21  E  cos  30°  cos  0  =  \/3  /  £  cc*  0 

Wl-W2          1 

*      w2  =  -j=r  tan  0,  as  above. 
IT 4+   1^2       v  3 


8          ALTERNATING  CURRENT  MOTORS. 

If  the  e.m.f.  of  A  B  be  not  equal  to  the  e.m.f.  of  A  C,  the 
reaching  of  the  wattmeter  in  either  position  may  be  corrected  to 
such  a  value  as  would  have  been  obtained  had  the  two  voltages 
been  equal,  in  which  case  the  above  relation  will  hold  true. 
•  Since  any  proportional  error  in  the  calibration  scale  of  the 
wattmeter  affects  equally  both  the  sum  and  the  difference  of 
the  readings,  and  hence  does  not  alter  the  ratio  of  the  two,  it 
follows  that  a  wattmeter  having  a  proportional  scale  error  of 


FIG.  8.— Testing  Circuits. 

any  value  whatever  may  be  used  to  obtain  the  correct  value 
of  lag  angle,  and  that  any  instrument  of  the  dynamometer  type, 
whether  calibrated  or  not,  may  be  used  in  place  of  the  wattmeter. 

The  lag  angle  thus  obtained  is  the  true  angle  between  the 
current  in  lead  A  X  and  the  mean  voltage  between  A  B  and  A  C. 

If  the  circuits  are  symmetrically  loaded  the  sum  of  two 
correct  wattmeter  readings  will  equal  the  real  power,  while 
from  the  difference  of  the  two  readings  may  be  obtained  the 
"  quadrature  "  watts. 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  9 

WATTMETERS  ON  UNBALANCED  LOADS. 

It  is  to  be  noted  especially  that  the  above  method  must  be  used 
with  care,  because  an  unbalance  of  load  may  lead  to  incorrect 
interpretation  of  the  results.  An  exaggerated  case  of  unbalance, 
selected  so  as  to  emphasize  the  facts  just  stated,  is  shown 
below.  In  order  that  all  disturbing  influences,  other  than  the 
unbalance  of  the  load  alone,  may  be  eliminated  from  the  prob- 
lem, there  has  been  assumed  a  non-inductive  load  supplied  from" 
a  three-phase  circuit  having  equal  e.m.fs.  between  the  leads. 

Fig.  8  represents  the  circuits  and  load,  and  shows  the  con- 


FIG.  9. — Vector  Diagram  of  Currents  and  Electromotive  Forces. 

nection  of  instruments  for  determining  the  power  factor.  As 
seen,  an  e.m.f.  of  100  volts  between  leads  and  a  delta-connected 
non-inductive  load  of  10,  20  and  30  amperes,  respectively,  per 
phase  have  been  chosen.  The  true  power  is  evidently  6,000 
watts,  while  the  power  factor  per  phase  is  unity. 

Referring  to  Fig.  9,  which  represents  the  value  and  position 
of  the  current  per  phase  of  the  load  indicated  by  Fig.  8,  the 
value  of  current  registered  upon  an  ammeter  at  C  and  its  rela- 
tive phase  position  may  be  ascertained  by  making  use  of  the 
geometrical  figure  c  d  }  e.  From  the  construction  it  is  seen  that 
the  current  at  c  is  represented  in  value  and  phase  by  the  line  c  f. 


10  ALTERNATING  CURRENT  MOTORS. 

In  the  triangle  c  e  /,  the  side  c  f  is  equal  to 

(c  e1  +  e~f  +  2  ~c~e  .  e  f  cos  120°)* 
or  is  equal  to 


(c  e*  +  c  e  ef  +  e  /*)*  =  V1900  =  43.60  and 
sin  <j>c  =  sin  120°  ^  =  .  866  |^  =     39?1 

,  C f  4d . DU 

<£<;  =  23°  24'. 
Similarly  the  current  at  a  is 

(102+10x  20  +  202)*  =  x/700  =  26.45,  and 

5*n00  -  -8662^  =  .3272 

<t>b  =  19°  6'. 
Current  at  6  is 

(102+10x  30  +  302)*  =  \/i300  =  36.05  and 
sinjb  =  -866^5  =  .2402 

OD .  UO 

fa  =  13°  54'. 

It  is  convenient  to  adopt  some  method  of  designating  at  once 
each  wattmeter  and  its  connection  in  the  circuit.  Place,  there- 
fore, as  subscript  to  the  letter  W,  which  is  to  represent  the 
reading  of  each  meter,  the  letters  showing  the  points  between 
which  the  voltage  coil  is  connected,  and  place  first  that  letter 
corresponding  to  the  lead  in  which  is  the  current  coil  of  the 
wattmeter.  Thus,  W  a  b  refers  to  wattmeter  having  its  current 
coil  in  lead  a  and  its  voltage  coil  connected  across  between  this 
lead  and  lead  6. 

Wattmeter  W  a  c  will  record  I  a  E  a  c  Cos  9  a  c,  or 
W  ac  =  26.45  X  100  X«w  19°  6'  =  2,500. 
W  a  b  -  26.45 X  100 Xcos  40°  54'  =  2,000. 
W  be  =  36.05 X  100 Xcos  13°  54'  =  3,500. 
Wba  =  36.05 X  100 Xow  46°  06'  =  2,500. 
Web  =  43.60XlOOXo?s  33°24'  =  4,000. 
Wca  -  43.60 X  100 Xo?s  36°  36'  =  3,500. 

It  is  seen  at  once  that  the  true  value  of  watts  is  recorded  in 
each  case  by  the  sum  of  the  readings  of  any  two  wattmeters 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  11 

with  their  current  coils  in  separate  leads  and  their  free  pressure 
terminals  connected  to  the  third  lead,  thus  (W  ac  +  Wbc}  = 
(W  ab  +  W  cb)  =  (Wba  +  W  ca)  =  6,000,  but  that  the  true 
watts  may  not  be  indicated  by  one  wattmeter  which  has  its 
pressure  coil  free  terminal  transferred  from  first  one  and  then 
the  other  remaining  lead,  thus  (W  ac  +  W  a  b}  =  4,500,  (W  b  c 
+  W  b  a)  =  6,000,  ( W  c  b  +  W  c  a)  =  7,500. 

It  was  shown  above  that  the  angle  of  lag  and  the  power 
factor  may  be  determined  by  the  ratio  of  the  readings  of  two 
wattmeters.  That  such  a  method  does  not  give  accurate  re- 
sults with  unbalanced  loads  was  mentioned  also.  For  purpose 
of  comparison,  however,  results  determined  by  this  method 
are  here  recorded.  Letting  0  represent  the  general  angle  of  lag, 

-Wbc-Wac  _  1000 

tan  6  --  ^  Wbc+Wac  =-  ^  6000  =  '2886 

e  =  16°  6' 

cos  6  =  .961 

.-Wcb-Wab  _2000 

tan °  --  - "  5772 


d  =  30°  0' 
cos  6  =  .866 

f£>«-viK5^'-*»!B-* 

0  =  16°  6' 
cos  6  =  .961 

Since  the  load  has  been  so  selected  as  to  be  strictly  non- 
inductive,  it  is  evident  that  the  lag  angle  indicated  does  not 
exist,  and  that  the  power  factor  obtained  by  this  method  is  in 
error. 

It  is  to  be  noted  in  this  connection,  however,  that  the  angle 
of  lag  obtained  by  the  same  formula  used  above,  but  sub- 
stituting the  ratio  of  readings  of  one  wattmeter  when  its  pressure 
coil  is  transferred  between  the  two  leads,  as  mentioned  above, 
has,  in  fact,  a  physical  significance,  as  here  shown: 

-W  ac-W  ab  _500 

iClM    \~i        ^~~     A  /   O   — •     *   /  O    —        1  QO^ 

V3Wac+Wab~'  V34500 
da  =  10°  45 


12  ALTERNATING  CURRENT  MOTORS.    ' 

An  inspection  of  Fig.  2  will  reveal  the  fact  that  this  is  the 
angle  between  the  current  at  a  and  the  mean  voltage  between 
ab  and  ac,  since  10°  54'  =  30°  -  (19°  +  6').  Similarly, 

-Wbc-Wba  1000 

tan  °b  =  ^  Wbc  +  Wba  -  ^3  6000 -  '2886 

Ob  =  16°.  6'  =  30°  -(13°.  54') 
and  again, 

-Wcb-Wca               500 
.  tan  uc  =  \f  3  j~ — - — — =  \/3  =  .1155 

6C  =  6°  36'  =  30°-  (23°.  24') 

A  popular  formula  for  determining  the  power  factor  of  a  three- 
phase  load  is 

w    .. 
P.P.  = 

where  I  is  the  current  per  lead  wire.     Substituting  the  values 
found  above  for  7,  the  power  factor  is 

_          6000          _ 
"  173.2X43.60  ~ 

6000 
"  173.2X36.05  " 

6000 


'-       178.6.45 

pF    =. 795+. 959+1. 309  =1Q21 

O 

Using  as  a  value  for  7  the  mean  current  per  lead  wire, 

6000          _. 
"  173.2X35.37 

Several  of  the  methods  used  above  are  obviously  in  great 
error,  and  their  use  would  never  be  sanctioned  in  a  careful  test. 
Few  objections,  however,  could  be  raised  against  the  last  two 
methods  of  averages,  though  neither  gives  the  true  result. 

In  the  determination  of  the  power  factor  as  the  ratio  of  true 
to  apparent  power,  the  question  arises  as  to  what  constitutes 
the  apparent  power,  and  the  discrepancies  in  results  are  due 
to  the  various  answers  which  may  be  given  to  this  question. 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  13 

While  doubt  must  ever  exist  as  to  the  value  to  be  assigned 
to  the  apparent  power  in  a  three-phase  system  operating  OQ  an 
unbalanced  load,  the  method  in  common  use  for  determining 
the  true  power  is  correct  for  any  condition  of  load,  proportion 
of  e.m.fs.  or  relation  of  power  factor  of  currents,  though  the 
methods  of  proof  of  this  fact,  which  are  based  on  assumptions 
of  equal  currents,  equal  power  factors,  or  equal  e.m.fs.  per 
phase,  are  evidently  open  to  many  objections. 

EQUIVALENT  SINGLE-PHASE  CURRENTS. 

Though  some  advantages  may  be  claimed  for  the  method  of 
dividing  the  amount  of  power  supplied  to  a  polyphase  motor  by 
the  number  of  phases  and  then  treating  the  machine  as  that 
number  of  single-phase  motors,  greater  simplicity  is  introduced 
into  the  calculation  if  equivalent  single-phase  qualities  be  de- 
termined for  the  polyphase  circuit.  In  accordance  with  this 
plan  the  total  number  of  watts  supplied  to  the  circuit  is  used 
in  computations  without  alteration,  the  measured  e.m.f.  of  the 
polyphase  circuit  is  considered  the  equivalent  single-phase 
e.m.f.,  while  the  equivalent  effective  current  is  understood  to 
mean  that  value  of  current  which  must  flow  at  the  same  power- 
factor  in  a  single-phase  circuit  at  the  same  voltage  to  transmit 
the  same  power. 

It  is  evident  that  in  a  two-phase  circuit  the  sum  of  the  cur- 
rents of  the  separate  phases  is  the  equivalent  single -phase 
current.  This  quantity  will  hereafter  be  referred  to  as  the 
"  total  current,"  or  as  simply  the  "  current  "  of  the  two-phase 
circuit.  Since  the  total  power  transmitted  in  a  three-phase 
system  is  expressed  by 

W  =  \/3  I  E  cos  6, 

v/3/  is  the  equivalent  effective  current.  In  the  equation 
above,  7  is  the  current  per  wire.  For  a  delta-connected  receiver 
the  current  per  phase  is  equal  to  7  -f-  \/3  or  the  sum  for  the  three 
phases  is  S/^v/S  =  \/37.  The  quantity  \/37  has,  therefore, 
a  physical  significance  as  the  total  current  in  a  delta-connected 
receiver,  though  in  a  star-connected  machine  such  quantity 
exists  only  mathematically.  However,  for  the  sake  of  com- 
bined generality  of  treatment  and  brevity  of  discussion,  x/§7 
will  hereafter  be  spoken  of  as  the  "  total  current,"  or  the  "  cur- 
rent "  or  the  "equivalent  single-phase  current"  of  the  three- 
phase  circuit. 


14 


ALTERNATING  CURRENT  MOTORS. 


EQUIVALENT  SINGLE-PHASE  RESISTANCE. 
In  determining  the  copper  loss  of  a  given  piece  of  polyphase 
apparatus,  it  is  convenient  to  know  what  value  of  resistance 
must  be  taken  so  that  the  square  of  the  total  current  may,  by 
its  multiplication  therewith,  give  the  actual  copper  loss  when 
the  corresponding  current  flows  in  the  circuit.  The  following 
very  simple  ratio  is  found  to  exist  between  the  resistance  of  a 


0000 

r 


FIGS.  10,  11  and  12. — Determination  of  Equivalent  Single- 
phase  Resistance. 


and 


given  circuit  as  measured  by  direct-current  instruments 
the  equivalent  resistance  for  the  total  current: 

For  any  two-phase  receiver  with  independent,  star,  three- 
wire,  or  mesh-connected  coils,  or  for  any  three-phase  receiver 
with  delta,  star  or  combination-connected  coils,  the  equivalent 
resistance  for  the  total  current  is  equal  to  one-half  of  the  value 
measured  between  phase  lines  by  direct-current  instruments. 


FIGS.  13,  14  and  15. — Determination  of  Equivalent  Single- 
phase  Resistance. 

The  proof  of  the  above  fact  for  circuits  connected  as  shown 
by  Figs.  10,  11  and  12,  is  self-evident  and  no  explanation  need 
be  given. 

Referring  to  Fig.  13  let  r  =  resistance  per  quarter;  then 
2  r  +  2  =  r  =  R  resistance  measured  by  direct-current  instru- 
ments. Let  i  =  current  per  coil ;  then  \/2  X  i  =  current  per 
lead  and  2  v^2  X i  =  total  current.  Evidently  the  copper  loss 


SINGLE-PHASE  AND  POLYPHASE  CIRCUITS.  15 

is  4  i2  r.  Using  the  total  current  as  above  and  R  -f-  2  as  the 
equivalent  resistance  the  copper  loss  is  (2  \/2  Xi)2Xr-r-2  =  4  i2  r. 
In  the  delta-connected  circuit  of  Fig.  14,  let  r  =  resistance  per 
coil ;  then  2  r  -j-  3  =  R  =  resistance  measured.  Let  i  =  current 
per  coil ;  then  3  i  —  total  current,  and  the  copper  loss  is  3  i2  r. 

For   equivalent   single-phase    quantities,    the   loss    is 

^  X  o 

=  3  i2  r. 

In  Fig.  15  let  r  =  resistance  per  coil;  then  2  r  =  R  =  resist- 
ance measured,  and  if  i  =  current  per  coil,  then  \/3  i  =  "  total  " 
current.  The  copper  loss  is  3  i2  r.  Using  "  total  "  current  and 
equivalent  resistance,  the  loss  is  (\/3  Xi)2r  =  3  i2  r. 

Since  the  ratio  of  effective  resistance  for  the  total  current 
to  the  measured  resistance  is  the  same  for  star  and  delta-connect- 
ed, three-phase  circuits,  the  same  ratio  must  evidently  hold  for 
a  combination  of  the  two  connections. 


CHAPTER  II. 
OUTLINE  OF  INDUCTION  MOTOR  PHENOMENA. 

METHODS  OF  TREATMENT. 

For  dealing  with  the  phenomena  of  induction  motors,  there  are 
numerous  points  from  which  the  problem  may  be  viewed,  each 
view  point  involving  a  certain  method  of  treatment,  but  it  may 
be  stated  that  in  general  all  methods  lead  to  practically  the 
same  results.  Thus  the  machine  may  be  treated  as  a  trans- 
former, or  it  may  be  considered  a  special  form  of  alternating- 
current  generator  delivering  current  to  a  fictitious  resistance 
as  a  load.  It  may  be  assumed  that  its  torque  is  due  to  the 
current  produced  in  the  secondary  of  the  transformer  of  one 
phase  acting  upon  the  magnetism  due  to  the  primary  of  another 
phase,  or  it  is  possible  to  consider  that  the  magnetisms  due  to 
the  separate  phases  combine  to  produce  a  revolving  field  in 
which  the  secondary  circuits  are  placed.  In  what  follows,  the 
induction  motor  will  be  looked  at  from  several  points  so  that  the 
reader  will  be  able  to  obtain  a  clear  view  of  the  actions  of  the 
machine,  a  systematic  attempt  being  made  to  present  the  motor 
in  such  a  light  as  to  avoid  all  unnecessary  complexities  which 
might  tend  to  blur  the  vision. 

Before  dealing  with  the  complex  inter-relation  of  the  com- 
ponent parts  of  the  motor  which  so  combine  in  their  actions  as 
to  produce  the  performance  obtained  from  the  structure,  it  is 
well  first  to  take  a  glance  at  the  machine  in  its  simplest  form 
so  as  to  see  just  what  may  be  expected  from  it.  It  is  believed 
that  this  method,  without  introducing  inaccuracies  incon- 
sistent with  the  object  sought,  possesses  the  advantage  of  allow- 
ing the  reader  to  become  quickly  acquainted  with  the  machine, 
and  permits  him  to  ascertain  the  involved  electromagnetic 
phenomena  and  to  study  them  singly,  and  then  conjointly,  in 
the  simplest  possible  manner. 

The  machine  that  will  be  discussed  here  is  the  ordinary  poly- 
phase induction  motor,  having  a  stationary  primary  wound 

16  - 


OUTLINE  OF  INDUCTION  MOTOR  PHENOMENA.         17 

with  overlapping  coils  in  slots,  and  a  revolving  secondary  which 
moves  in  the  rotating  field  set  up  by  the  primary  windings. 
It  may  be  well  at  this  point  to  call  attention  to  the  fact  that 
the  primary  coils  can  occupy  either  the  moving  or  the  stationary 
member,  provided  that  the  stationary  or  the  moving  member, 
respectively,  is  wound  with  the  secondary  coils.  Due  to  this 
fact  the  terms  "  armature  "  and  "  field  "  members,  when  ap- 
plied to  an  induction  motor,  are  apt  to  lead  to  confusion.  It 
should  be  noted  that  the  machine  has  two  windings,  the  li  pri- 
mary "  and  the  "  secondary,"  either  of  which  may  be  placed 
on  the  "  rotor  "  or  the  "  stator." 

PRODUCTION  OF  REVOLVING  FIELD. 

As  usually  built,  the  primary  coils  form  a  distributed  winding 
similar  to  that  of  a  direct  current  armature.  These  coils  are 
connected  into  groups  according  to  the  number  of  phases 
and  poles  for  which  the  machine  is  designed,  there  being  one 
group  per  phase  per  pole. 

The  current  for  each  phase  is  run  through  the  corresponding 
group  of  each  pole,  tending  to  make  alternate  north  and  south 
magnetic  poles  around  the  machine.  These  north  and  south 
poles  of  each  phase  combine  with  those  of  the  other  phases  to 
make  resultant  north  and  south  poles.  These  resultant  poles 
shift  positions  with  the  alternations  of  the  currents  and  occupy 
places  on  the  primary  core  determined  by  the  relative  strengths 
of  the  currents  in  the  different  groups  of  coils.  .Each  magnetic 
pole,  therefore,  advances  by  the  arc  occupied  by  one  group  of 
windings  for  each  phase  during  each  alternation  of  the  current. 
It  is  thus  seen  that  the  resultant  magnetic  field  revolves  at  a 
speed  depending  directly  upon  the  alternations  and  inversely 
upon  the  number  of  poles. 

In  the  revolving  field  is  situated  the  secondary,  in  the  windings 
of  which  is  generated  an  e.m.f.  determined  by  the  rate  at  which 
its  conductors  are  cut  by  the  magnetic  lines  from  the  primary. 
If  the  circuits  of  the  secondary  be  closed  there  will  flow  therein 
a  current  which,  being  in  a  magnetic  field,  will  exert  a  torque 
tending  to  cause  the  secondary  member  to  turn  in  the  direction 
of  the  revolving  field.  The  final  result  is  that  the  speed  of 
the  rotor  increases  to  a  value  such  that  the  relative  motion  of 
the  secondary  conductors  and  the  revolving  field  generates 


18  ALTERNATING  CURRENT  MOTORS. 

an  e.m.f.  sufficient  to  cause  to  flow  through  the  impedance  of 
the  conductors  a  current,  the  product  of  which  into  the  strength 
of  the  field  will  equal  the  torque  demanded. 

As  can  be  seen,  the  speed  of  the  rotor  can  never  equal  the 
speed  of  the  rotating  magnetic  field,  because  the  conductors  of 
the  secondary  must  cut  the  lines  of  force  of  the  field  in  order 
to  cause  current  to  flow  in  the  secondary  winding,  and  pull  the 
rotor  around;  if  the  speeds  were  identical  there  would  be  no 
cutting.  The  difference  between  these  two  speeds  is  commonly 
known  as  the  "  slip."  For  the  purpose  of  this  discussion  the 
slip  will  be  considered  in  terms  of  the  synchronous  speed,  that 
is  to  say,  if  the  synchronous  speed  were  1200  r.p.m.  and  the 
actual  speed  of  the  rotor  or  secondary  member  were  1176  r.p.m., 
the  slip  would  be  24-- 1200  =  0.02. 

Since  the  conductors  of  the  secondary  cut  an  alternating  field, 
first  of  one  polarity  and  then  of  the  other,  the  current  produced 
in  them  will  be  alternating.  The  frequency  of  this  current 
is  normally  much  lower  than  the  frequency  of  the  supply  cur- 
rent ;  the  secondary  frequency  is  equal  to  s  f,  s  being  the  slip 
and  ;  the  frequency.  The  current  in  the  secondary  being  alter- 
nating, the  impedance  of  the  secondary  winding  has  a  reactive 
component  in  addition  to  its  resistance.  The  reactance  at 
standstill  is,  of  course,  equal  to 

2  TT  /  L2  =  X2. 

/  being  the  frequency  of  the  supply  current  and  L2  the  coefficient 
of  self-induction  of  the  secondary  winding.  The  secondary  re- 
actance when  the  marhine  is  in  operation  is  equal  to 

2xfL2s  =  s  X2, 

s  being  the  slip,  as  previously  explained. 

It  is  evident  that  the  reactance  of  the  secondary  increases 
directly  with  the  slip,  so  that  with  a  constant  value  of  secondary 
resistance  the  impedance  increases  as  the  slip  increases;  not  in 
direct  proportion,  however.  The  effect  of  the  reactive  com- 
ponent of  the  secondary  impedance  is  to  cause  the  secondary 
current  to  lag  behind  the  secondary  e.m.f.  by  an  "  angle,"  the 
cosine  of  which  is  equal  to  the  resistance  of  the  circuit  divided 
by  its  impedance.  If  6  represents  this  angle,  cos  6  represents 
the  power  factor  of  the  secondary  current. 

Looking  further  into  the  effect  of  increasing  the  load  on  the 


OUTLINE  OF  INDUCTION  MOTOR  PHENOMENA.         19 

motor,  it  will  be  plain  that  when  the  rotor  is  running  near 
synchronism  the  reactance  is  of  negligible  effect,  so  that  the 
secondary  current  increases  directly  with  the  slip  and  has  a 
power  factor  of  practically  unity.  As  the  load  increases,  the 
torque  must  be  greater;  the  motor  slip  therefore  increases,  with 
a  consequent  increase  of  secondary  e.m.f.  As  the  reactance 
begins  now  to  increase,  the  impedance  also  increases,  and  the 
secondary  current  is  no  longer  proportional  directly  to  the 
secondary  e.m.f.,  and  the  current  which  does  flow  has  a  power 
factor  less  than  unity,  that  is  to  say,  it  is  out  of  phase  with  the 
secondary  e.m.f.  Thus  it  is  out  of  time  phase  with  the  revolving 
magnetism,  thereby  requiring  a  proportionately  larger  current 
to  produce  a  corresponding  torque. 

The  increased  draft  of  current  demanded  for  the  primary 
circuit  entails  a  drop  of  e.m.f.  in  the  primary  windings,  and, 
since  there  must  be  mechanical  clearance  between  the  primary 
and  secondary  cores,  the  lines  of  force  which  pass  from  the 
primary  to  the  secondary  are  diminished  by  the  increase  of  the 
secondary  current,  which  always  flows  so  as  to  oppose  the  mag- 
netism causing  it.  Therefore  the  magnetic  field,  which  produces 
the  secondary  e.m.f.,  decreases  with  increase  of  load.  Since 
only  the  qualitative  behavior  of  induction  motors  is  necessary 
for  the  purpose  of  this  discussion,  and  no  regard  need  be  had 
at  present  for  the  quantitative  performance,  this  effect  will 
be  momentarily  neglected. 

SIMPLE  ANALYTICAL  EQUATIONS. 

Let      E2  =  secondary  e.m.f.   at   standstill; 
s  E2  =  secondary  e.m.f.  at  a  slip  s; 

X2  =  secondary  reactance  at  standstill; 
s  X2  =  secondary  reactance  at  a  slip  5  ; 

R2  =  secondary  resistance. 
then  \/R22  +  s2  X22  =  secondary  impedance, 

s  E  R 

I  =      .  2=  =  secondary  current,  and  cos  6  = 


=    secondary  power  factor;  all  three  at  the  slip,  s. 

D  =  v*.'+!»  x.'VaA  xf  x*=secondary  to^  *  ^e 

slip,  s;  where  K  is  a  constant  depending  upon  the  terms  in 


20  ALTERNATING  CURRENT  MOTORS. 

which  torque  is  expressed,  and  upon  the  magnetic  field,  —  the 
latter  being  here  assumed  constant. 

/?      o    J7      fc" 

Hence  D  =  r>22  ,    I  V22  =  rotor  torque  at  the  slip  5. 

/V2      i    ^     A  2 

An  examination  of  these  formulas  reveals  many  of  the  char- 
acteristics of  the  induction  motor,  which  it  is  well  to  discuss 
at  this  point. 

(1)  The  torque  becomes  maximum  when  R2  =  s  X2. 

(2)  If  R2  be  made  equal  to  X2,  maximum  torque  will  occur 
at  standstill. 

(3)  Since  at  maximum  torque  R2  =  5  X2  the  torque  is  equal  to 

s2X2E2K  =  E2K 
2  s2  X}       ~  2  X2  ' 

and  the  value  of  maximum  torque  is  independent  of  the  resist- 
ance. 

5  E 

(4)  When    X2    is    negligible    the    torque    =    K  -5-*,   or  the 

K2 

torque  is  directly  proportional  to  the  slip  near  synchronism, 
and  inversely  to  the  resistance. 

PC  7?    /** 

(5)  At  standstill   the   torque  =  ~^-  —  ^r-2  which,   when  R2  is 

K2    +   A2 

less  than  X2  can  be  increased  by  the  insertion  of  resistance  up 
to  the  point  where  the  two  are  equal;  further  increase  of  resist- 
ance will  then  decrease  the  torque. 

(6)  The  starting  torque  is  proportional  to  the  resistance  and 
inversely  proportional  to  the  square  of  the  impedance. 

(7)  Since  power  is  proportional  to  the  product  of  speed  and 
torque,  the  output  is  equal  to 

TV-   D     o    77 

P  =  A  (1  —  s)          2      v2  2  =  A   (1  —  s)  D,  and  is  a  maximum 
K2  -\-s~  A  2 

at  a  slip  less  than  that  giving  maximum  torque. 


and  at  maximum  torque  R2    =  s2  X22.     Therefore,  7  at  maximum 

Z7*  Z7* 

torque  =      ._    2..  =      ._.  2    ;  whence  it  is  plain  that  the  secondary 
V2  s  X2      V  2  A 

current  giving  maximum  torque  is  independent  of  the  secondary 
resistance. 


OUTLINE  OF  INDUCTION  MOTOR  PHENOMENA.          21 

(9)  At  maximum  torque  the  secondary  power  factor  = 

0.707. 

2  £2  R         5  E2D 
K 


(10)  PR,  =      *      22  *l    =        »        and  since  E2  is  constant 


for  constant  impressed  pressure,  the  copper  loss  of  the  secondary 
is  proportional  to  the  product  of  the  slip  and  torque. 

(11)  For  a  given  torque  the  slip  is  proportional  to  the  copper 
loss  of  the  secondary  and  independent  of  the  secondary  reactance 
or  coefficient  of  self-induction. 

(12)  Output  is  equal  to  P  =  I  E2  (1  -  5)  cos  0 

P  = 


X22 


(13)    If  all  losses   except   that   of   the    secondary   copper   be 
neglected,  the  input  will  be 


.    V  •>  jrr       **"'  2         V  /  '  TS" 

A.  /v 

and  the  efficiency  will  be 


which  means  that  the  efficiency  is  equal  to  the  absolute  speed 
in  per  cent,  of  synchronism.  Since  there  must  be  losses  in 
addition  to  that  of  the  secondary  copper,  the  efficiency  is  always 
less  than  the  speed  in  per  cent,  of  synchronism. 

(14)  At  a  given  slip  the  torque  varies  as  the  square  of  the 
primary  pressure.  This  is  seen  from  the  fact  that  the  sec- 
ondary current  at  a  given  slip  will  vary  directly  as  the  strength 
of  field,  the  power  factor  will  remain  constant,  and  therefore  the 
torque  which  is  obtained  from  the  product  of  secondary  current, 
power:  factor  and  field  will  vary  as  the  square  of  the  field.  Since 
the  strength  of  the  field  varies  directly  with  the  primary  pressure 
at  a  given  slip,  the  torque  will  vary  as  the  square  of  the  primary 
pressure. 


22  ALTERNATING  CURRENT  MOTORS 

STARTING  DEVICES  FOR  INDUCTION  MOTORS. 

Having  thus  determined  the  behavior  of  the  induction  motor 
under  various  changes  of  condition,  the  next  step  is  to  ascertain 
the  methods  by  which  it  can  be  adapted  to  services  of  different 
requirements. 

If  the  impedance  of  the  secondary  be  sufficiently  great  to 
prevent  a  destructive  flow  of  current  when  full  pressure  is  ap- 
plied to  the  primary  with  the  rotor  at  standstill,  the  motor 
may  be  started  by  being  connected  directly  to  the  supply  cir- 
cuit. This  is  a  method  adopted  quite  extensively  for  motors 
of  small  size.  If  the  impedance  consists  for  the  most  part  of 
reactance,  the  current  drawn  will  have  a  low  power  factor, 
which,  in  general,  affects  materially  the  regulation  of  the  sys- 
tem ;  the  starting  torque  will  be  small. 

It  was  proved  above  that  by  so  proportioning  .the  secondary 
resistance  that  R2  =  X2  the  maximum  torque  would  be  exerted 
at  standstill.  With  resistance  of  such  a  value  permanently 
connected  in  the  secondary  circuit,  at  full  load  the  slip  would  be 
enormous  and  the  efficiency  correspondingly  low,  as  shown 
above.  As  a  compromise  in  this  respect,  motors  are  usually 
constructed  with  only  comparatively  large  resistance  in  the 
secondary  windings. 

In  order  to  reduce  the  current  at  starting,  large  motors  are 
often  arranged  to  be  supplied  with  a  lower  primary  e.m.f.  and 
the  full  line  e.m.f.  applied  only  after  they  have  attained  a  fair 
speed.  This  is  also  especially  desirable  when  otherwise  the 
excessive  starting  torque  would  produce  mechanical  injury  to  the 
shafting,  etc.,  driven  by  the  motor.  For  elevators  and  cranes 
this  method  has  found  extensive  application.  Crane  motors 
are  built  with  a  secondary  resistance  which  gives  maximum 
torque  at  standstill.  The  reduced  primary  pressure  is  secured 
either  from  lowering  transformers  or  from  compensators  (auto- 
transformers).  In  either  case,  loops  are  taken  to  a  suitable 
controller,  by  which  the  pressure  supplied  to  the  primary  of  the 
motor  is  governed.  Pressures  of  a  number  of  different  values 
are  thus  obtained.  The  very  intermittent  character  of  the 
work  performed  by  crane  motors  renders  their  low  efficiency  a 
necessary  evil,  since  in  any  case  the  efficiency  must  be  less  than 
the  speed  in  per  cent,  of  synchronism. 

In  order  to  combine  large  starting  torque  with  good  running 


OUTLINE  OF  INDUCTION  MOTOR  PHENOMENA.          23 

speed  and  efficiency,  it  is  customary  to  provide  resistance  ex- 
ternal to  the  secondary  windings  and  arrange  to  suitably  reduce 
this  resistance  as  the  speed  increases,  allowing  the  motor  to 
run  without  external  resistance  for  the  highest  speed.  For 
continuous  service  this  method  has  proved  highly  satisfactory 
and  gives  the  best  running  efficiency.  Ordinarily,  this  necessi- 
tates the  use  of  collector  rings  for  the  revolving  member,  whether 
such  be  the  primary  or  the  secondary.  When  the  secondary 
revolves  it  is  universally  given  a  three-phase  winding,  inde- 
pendent of  the  number  of  phases  of  the  primary,  and  therefore 
three  collector  rings  are  used.  The  external  resistance  is  con- 
nected either  "  delta  "  or  "  star,"  and  regulated  by  a  suitable 
controller. 

A  very  ingenious  device  for  automatically  adjusting  the  ex- 
ternal resistance  was  exhibited  at  the  Paris  Exposition.  It 
was  shown  above  that  the  frequency  of  the  secondary  e.m.f. 
depends  directly  upon  the  slip  and  is  a  maximum  at  standstill. 
If  an  induction  coil  of  low  resistance  and  high  inductance  be 
subjected  to  an  e.m.f.  of  varying  frequency,  the  admittance 
will  vary  inversely  with  the  frequency.  In  the  Fischer-Hinnen 
device,  just  referred  to,  there  is  connected  external  to  the  sec- 
ondary windings  a  non-inductive  resistance  of  a  value  to  give 
practically  maximum  torque  at  standstill.  In  parallel  with 
this  is  connected  a  highly  inductive  coil  of  extremely  small 
resistance.  At  zero  speed  the  admittance  of  the  external  im- 
pedance consists  practically  of  only  that  of  the  non-inductive 
resistance.  As  the  speed  increases  the  admittance  increases, 
due  to  the  decreased  reactance,  and  at  synchronism  the  external 
impedance  is  somewhat  less  than  the  resistance  of  the  inductive 
coil.  The  action  is,  therefore,  in  effect  the  same  as  though  the 
external  resistance  had  been  decreased  directly  as  the  speed 
increased.  It  is  claimed  that  the  starting  device  occupies  so 
small  a  space  that  it  can  be  placed  in  the  rotating  armature, 
so  that  no  collector  rings  are  required. 

CONCATENATION  CONTROL. 

A  common  defect  in  all  methods  of  speed  regulation  thus  far 
described  is  the  low  efficiency  at  speeds  far  below  synchronism. 
Attention  has  been  frequently  directed  to  the  fact  that  the  effi- 
ciency is  in  any  case  less  than  the  speed  in  per  cent,  of  syn- 


24  ALTERNATING  CURRENT  MOTORS. 

chronism.  In  order,  therefore,  to  run  at  high  efficiency  at 
reduced  speed,  it  is  necessary  to  adopt  some  method  of  de- 
creasing the  synchronous  speed. 

The  synchronous  speed  of  a  motor  is  equal  to  the  alternations 
of  the  system  divided  by  the  number  of  poles  of  the  motor. 
If,  therefore,  a  motor  is  arranged  for  two  different  numbers  of 
poles  it  will  have  two  different  synchronous  speeds.  While 
offering  many  advantages  over  other  more  complicated  systems 
of  speed  control,  and  introducing  no  unsurmountable  difficulties 
in  practical  application  to  existing  types  of  motors,  this  method 
has  as  yet  passed  little  beyond  the  stage  of  experimental  in- 
vestigation. 

With  mechanical  connection  between  two  motors,  "  con- 
catenation "  or  "  tandem  "  control  also  offers  a  means  of  reducing 
the  synchronous  speed.  In  practice,  the  secondary  of  one  motor 
is  connected  to  the  primary  of  the  other,  the  primary  of  the 
first  being  connected  to  the  line.  The  frequency  of  the  current 
in  the  secondary  of  any  motor  depends  upon  its  slip,  as  noted 
above.  At  standstill,  therefore,  the  frequency  impressed  upon 
the  primary  of  the  second  motor  will  be  that  of  the  line.  As 
the  motor  increases  in  speed  the  frequency  of  the  secondary 
of  the  first  motor  decreases,  and  at  half  speed  the  frequency 
impressed  upon  the  primary  of  the  second  motor  will  be  equal 
to  the  speed  of  its  secondary ;  that  is,  it  will  have  reached  its 
synchronous  speed.  If  now,  the  primary  of  the  second  motor 
be  connected  to  the  supply  circuit  and  the  secondary  of  the  first 
motor  be  connected  to  a  resistance,  the  motors  will  tend  to  in- 
crease in  speed  up  to  the  full  synchronism  of  the  supply. 

By  the  use  of  suitably  selected  resistances  for  the  secondary 
circuits  of  the  two  motors,  this  method  gives  the  same  results 
as  the  series-parallel  control  of  direct-current  series  motors,  with 
one  important  distinction,  however.  The  series- wound  motors 
tend  to  increase  indefinitely  in  speed  as  the  torque  is  diminished, 
while  the  induction  motors  tend  to  reach  a  certain  definite  speed, 
above  which  they  act  as  generators.  In  this  respect  they  re- 
semble quite  closely  two  shunt  motors  with  constant  field  ex- 
citation, having  armatures  connected  successively  in  series  and 
in  parallel,  with  and  without  resistance,  and,  like  the  shunt 
motors,  when  driven  above  the  normal  speed  they  feed  power 
back  to  the  supply. 


CHAPTER  III. 
OBSERVED  PERFORMANCE  OF  INDUCTION  MOTOR. 

TEST  WITH  ONE  VOLTMETER  AND  ONE  WATTMETER. 

In  the  preceding  chapter,  a  hasty  glance  was  taken  at  the 
characteristics  of  an  induction  motor  under  certain  assumed 
ideal  conditions.  It  is  well  to  show  from  actual  tests  just  how 
closely  the  observed  results  compare  with  the  ideal  calculated 
results.  Below  there  is  given,  therefore,  the  complete  test  of 


FIG.   16. — Test  of  Three-phase  Motor  with  One  Voltmeter 
and  one  Ammeter. 


an  induction  motor  under  actual  operating  conditions,  and  in- 
cidentally mention  is  made  of  a  convenient  method  for  testing 
a  motor  when  the  supply  of  instruments  is  limited. 

The  method  of  testing  a  transformer  or  a  generator  by  sep- 
aration of  the  losses  is  well  known.  It  is  such  a  method,  which 
by  a  few  slight  modifications  can  be  applied  to  induction  motors, 
that  is  given  below.  Most  of  what  is  stated  in  this  connection 
is  true  for  any  induction  motor  under  any  condition  of  service, 

25 


26  ALTERNATING  CURRENT  MOTORS. 

though  the  greatest  simplicity  in  testing  and  the  requisite  use 
of  the  least  number  of  instruments  will  be  obtained  only  with 
polyphase  motors  operating  on  well  balanced  and  regulated 
circuits.  Each  element  of  a  test  is  treated  separately. 

MEASUREMENT  OF  SLIP. 

The  determination  of  the  slip  of  induction  motor  rotors 
by  counting  the  r.p.m.  of  both  generator  and  motor  is 
open  to  many  objections.  If  the  two  readings  of  speed  be  not 
taken  simultaneously  though  the  true  value  of  each  be  correctly 
observed,  when  the  speed  of  either  is  fluctuating  the  value  of 
slips  will  be  greatly  in  error.  A  slight  proportional  error  in 
either  speed  introduces  an  enormous  error  in  the  slip.  Where 
the  generator  is  not  at  hand  the  above  method  obviously  cannot 
be  directly  applied,  and  is  applicable  only  when  a  synchronous 
motor  is  available  for  operation  from  the  same  supply  system 
as  the  induction  motor. 

When  the  secondary  current  can  be  measured  and  the  secondary 
resistance  is  known,  the  most  accurate  and  convenient  method 
for  determining  the  slip  is  from  the  ratio  of  copper  loss  of  sec- 
ondary to  total  secondary  input. 

If  we  let  7  2  be  any  observed  value  of  secondary  current,  R2 
the  secondary  resistance  and  W0  the  output  of  the  motor,  then 

722  R2  Copper  loss  of  secondary 

W0  +  7  2  R2         total  secondary  input 

As  a  proof  of  this  fact,  consider  the  magnetism  cut  by  the 
secondary  windings  to  be  of  a  strength  which  would  cause  to 
be  generated  E2  volts  in  the  windings  at  100  per  cent.  slip. 

Let  5  be  any  given  slip,  Ws  the  total  secondary  watts,  6  the 
angle  of  lag  of  secondary  current,  and  Xz  the  secondary  re- 
actance at  100  per  cent,  slip,  then 


W5 


sec  6 


722^2  SI2E2R2 

72  E2  cos  0       72  £2  R2  cos  0  sec  0 


OBSERVED  PERFORMANCE  OF  INDUCTION  MOTOR.     27 

Since  neither  E2  nor  X2  appears  in  the  above  equation,  the 
relation  is  independent  of  the  strength  of  field  magnetism  cut 
by  the  secondary  windings  and  of  the  secondary  reactance. 

DETERMINATION  OF  TORQUE. 

The  torque  of  the  rotor  can  be  ascertained  with  the 
highest  degree  of  accuracy  and  facility  from  the  ratio  of  sec- 
ondary input  to  the  synchronous  speed,  that  is,  the  torque  is 
expressed  in  pounds  at  one-foot  radius  by  the  following  equa- 
tion: 

W    -  T 

Torque  =  D  =  7.04  -^-  =*^ 
syn.  speed 

where  WP  is  the  total  primary  input;  LP  the  total  primary 
losses,  and  the  synchronous  speed  is  in  r.p.m. 

If  LP  includes  the  friction,  D  will  equal  the  external  rotor 
torque,  while  if  LP  includes  only  the  true  primary  iron  and 
copper  losses,  D  will  be  the  total  rotor  torque. 

To  prove  the  above  expressed  relation,  let  Ws  be  total  sec- 
ondary input,  W0  the  motor  output,  and  s  the  rotor  slip.  Then 

=     33  OOP  W0 
2  n  r.p.m.  746' 

but  WQ  =  W  s  (1  — 5)  and  r.  p.  m.  =  synchronous  speed  (1  — s) . 
Therefore, 

Ws 


D  =  7.04 


syn.  speed' 


Therefore,  for  a  given  primary  input  and  primary  losses  the 
rotor  torque  is  independent  of  secondary  speed  or  output,  and 
any  error  in  the  determination  of  either  of  the  latter  quantities 
need  not  affect  in  the  least  the  value  obtained  for  the  torque. 

If  the  total  power  received  by  the  secondary  is  used  up  in  the 
secondary  resistance  the  equation  for  the  torque  will  be 

D0  =  7.04      /z2  ^2    . 
syn.  speed 

or  starting  torque,  which,  as  has  been  stated  previc  :sly,  may  be 
increased  by  any  method  which  will  increase  the  stationary 
secondary  copper  losses. 


28  ALTERNATING  CURRENT  MOTORS. 

MEASUREMENT  OF  SECONDARY  RESISTANCE. 

Due  to  the  inability  to  insert  measuring  instruments  in  the 
secondary  of  squirrel-cage  induction  motors,  the  ordinary 
direct-current  method  of  determining  the  resistance  cannot  be 
used. 

If  the  rotor  be  clamped  to  prevent  motion  a  wattmeter 
placed  in  the  primary  circuit  will  read  the  copper  loss  of  both 
primary  and  secondary  when  the  e.m.f.  across  the  leads  is  re- 
duced to  give  a  fair  operating  value  of  primary  current.  If 
from  the  reading  of  watts  thus  obtained  there  be  subtracted 
the  known  copper  loss  of  the  primary  for  the  current  flowing, 
the  value  of  the  secondary  copper  loss  will  be  secured.  The 
resistance  of  the  secondary  (reduced  to  primary)  will  be  ob- 
tained by  dividing  this  loss  by  the  square  of  the  primary  current. 
It  is  to  be  noted  that  certain  minor  effects  are  here  neglected. 
These  effects  are  of  small  moment  and  do  not  seriously  modify 
the  final  results.  However,  they  will  be  treated  at  length  in  a 
later  chapter. 

DETERMINATION  OF  SECONDARY  CURRENT. 

When  the  motor  is  provided  with  a  squirrel  cage  secondary 
winding,  measurement  of  the  secondary  current  cannot  be 
made  directly,  but  the  determination  of  its  value  must  be 
by  calculation.  The  variation  in  value  and  phase  of  the 
primary  current  may  serve  as  an  indication  of  the  current 
flowing  in  the  secondary  windings,  though  the  actual  increase  in 
primary  current  does  not  represent  the  increase  in  secondary 
current. 

The  difference  between  the  power  components  of  the  primary 
current  at  no  load  and  under  a  chosen  load  may  be  taken  as  the 
equivalent  increase  in  the  power  component  of  the  secondary 
current  under  the  same  conditions,  while  the  difference  between 
the  quadrature  components  of  the  primary  current  at  the  same 
time  is  a  measure  of  the  equivalent  increase  in  the  quadrature 
component  of  the  secondary  current. 

When  the  no-load  value  of  secondary  current  is  negligible 
the  vector  sum  of  the  above-found  components  represents 
the  secondary  current  for  the  chosen  load  in  terms  of  the 
primary  current.  These  facts  will  be  discussed  more  fully 
hereafter. 


OBSERVED  PERFORMANCE  OF  INDUCTION  MOTOR.     29 

CALCULATION  OF  PRIMARY  POWER  FACTOR. 

For  a  single-phase  motor  the  determination  of  the  primary 
power  factor  will  usually  involve  the  measurement  of  primary 
watts,  volts  and  amperes. 


5  6 

Horse  Power  Output 

FIG.  17. — Test  Characteristics  of  Induction  Motor. 

For  two-phase  motors  with  equal  e.m.fs.  across  the  separate 
phases,  one  wattmeter  alone  may  be  used  to  obtain  the  power 
factor  by  simply  transferring  the  pressure  coil  from  one  phase 


30  ALTERNATING  CURRENT  MOTORS. 

to  the  other,  leaving  the  current  coil  always  in  one  lead.     The 
reading  of  the  wattmeter  in  one  case  will  be  Wl  —  I  E  cos  6, 

and  in  the  second  case,   W2  —  /  E  sin  0\  whence  tan  6  =  —  2 

from  which  may  be  obtained  the  power  factor. 

For  three-phase  motors  one  wattmeter  can  similarly  serve  to 
indicate  the  power  factor.  The  wattmeter  readings  will  be 
Wl  =  /  E  cos  (0  -  30)  ;  W2  =  7  E  cos  (0  +  30), 

_ 
whence  tan0  ==  >/3 


CALCULATION  OF  PRIMARY  CURRENT. 

If  the  primary  electromotive  force  watts,  and  power  factor 
be  known,  the  primary  current  can  readily  be  calculated  and, 
therefore,  need  not  be  measured. 

It  is  evident  from  the  above  discussed  facts  that  with  two 
and  three-phase  induction  motors  operated  from  circuits  having 
constant  and  equal  e.m.fs.  across  the  separate  phases,  one  watt- 
meter and  one  voltmeter  can  be  used  to  determine  primary 
watts,  amperes  and  volts,  and  secondary  amperes,  and  that 
when  the  primary  resistance  is  known  or  can  be  measured  the 
complete  performance  efficiency,  etc.,  of  the  motors  can  at  once 
be  calculated. 

In  the  above  table  and  in  Fig.  17  there  are  given  the  calculations 
and  the  curves  of  such  a  test  made  upon  a  5-h.p,,  eight-pole, 
60-cycle,  three-phase  induction  motor.  The  primary  resistance 
between  leads  at  running  temperature  was  .156  ohm,  and  the 
equivalent  secondary  resistance  (found  as  above)  was  .56  ohm. 

Since  the  copper  loss  of  a  three-phase  receiver  is  expressed  by 

the  equation,  ,*/  -    R  P,   where   R  is   the   resistance   between 

lines  and  7  is  the  current  flowing  in  each  lead,  for  either  star 
or  delta-connected  receiver  no  attention  need  be  paid  to  the 
method  by  which  the  primary  coils  are  interconnected  within 
the  motor  or,  in  fact,  whether  the  secondary  be  wound  delta, 
star  or  squirrel  cage. 

Tests  made  upon  this  same  motor  by  the  output-input  meth- 
ods agree  throughout  the  whole  range  of  the  test  with  the 
herewith  recorded  test  within  the  limits  of  the  inevitable  errors 
of  observation  of  the  various  instruments  used  in  the  tests. 


CHAPTER  IV. 
INDUCTION  MOTORS  AS  FREQUENCY  CONVERTERS. 

FIELD  OF  APPLICATION. 

For  the  satisfactory  operation  of  arc  lamps  a  frequency  higher 
than  40  p.p.s.  is  required,  while  when  the  frequency  is  much 
below  this  value  incandescent  lamps  may  show  a  fluctuation 
in  brilliancy.  The  output  of  transformers  may  be  shown  to 
vary  as  the  three-eighth  power  of  the  frequency.  These  are 
among  the  causes  for  the  fact  that  in  the  older  lighting  stations 
a  frequency  as  high  as  133  p.p.s.  was  quite  common. 

With  later  increase  of  magnitude  and  range  of  service,  it  was 
found  that  a  lower  frequency  improved  the  operation  of  alter- 
nators in  parallel,  while  the  line  regulation  was  also  bene fitted 
by  the  change  from  the  higher  frequency.  This  led  to  the  adop- 
tion of  a  periodicity  of  about  50  to  60  p.p.s.  With  the  advent 
of  the  long-distance  power  transmission  circuits,  the  advisability 
of  the  adoption  of  a  still  lower  frequency  became  apparent, 
while  the  successful  use  of  rotary  converters  for  railway  work 
practically  necessitates  a  frequency  as  low  as  25  cycles.  This 
is  the  present  standard  frequency  for  such  service. 

When  lighting  is  to  be  done  from  power  supplied  at  25  cycles, 
some  means  must  be  provided  for  altering  the  nature  of  the 
current  before  it  can  be  applied  to  the  lamps.  A  most  satis- 
factory method  of  accomplishing  this  result  is  by  means  of  alter- 
nating-current motors,  of  either  the  induction  or  synchronous 
type,  driving  lighting  generators.  Where  only  high- voltage 
alternating  currents  are  available  this  method  requires  the  use 
of  step-down  transformers,  a  motor  and  a  generator,  each 
carrying  the  full  load.  When  the  pressure  at  hand  is  suffi- 
ciently low,  step-down  transformers  may  be  dispensed  with, 
but  a  double  equipment  and  double  transformation  of  power 
is  still  necessary,  with  consequent  low  efficiency  and  high  cost 
of  installation. 

A  convenient  method  for  changing  the  lower  frequency  to  a 

31 


32  ALTERNATING  CURRENT  MOTORS. 

value  suitable  for  lighting  purposes  is  by  the  use  of  "frequency 
converters,"  which  constitute  a  special  adaptation  of  induction 
motors  as  secondary  circuit  generators. 

CHARACTERISTIC  PERFORMANCE. 

In  the  ordinary  induction  motor  the  frequency  of  the  sec- 
ondary current  is  not  that  of  the  supply,  but  it  has  a  value 
represented  by  the  product  of  the  slip  of  the  rotor  from  syn- 
chronous speed  and  the  frequency  of  the  primary  current. 
It  is  only  when  the  slip  is  unity,  or  at  standstill  that  the  pri- 
mary and  secondary  frequencies  are  equal.  Under  this  con- 
dition the  windings  are  in  a  true  static  transformer  relation, 
and,  with  the  rotor  clamped  to  prevent  relative  motion,  current 
at  the  primary  frequency  can  be  drawn  from  the  secondary 
windings.  Obviously,  the  air-gap  renders  the  induction  motor 
for  such  purposes  much  inferior  to  a  static  transformer,  on 
account  of  magnetic  leakage  between  the  coils. 

If,  now,  the  secondary  be  given  a  motion  relative  to  the 
primary,  there  may  continue  to  be  drawn  from  the  secondary, 
current  at  a  frequency  determined  by  the  slip  from  synchronous 
speed.  If  this  slip  be  greater  than  unity — that  is,  if  the  motor 
be  driven  backwards — the  frequency  of  the  secondary  current 
will  be  greater  than  that  of  the  primary.  By  properly  propor- 
tioning the  rate  of  backward  driving,  the  secondary  current 
can  be  given  a  frequency  of  any  desired  value. 

In  order  that  the  secondary  frequency  may  bear  a  constant 
ratio  to  the  primary,  it  is  necessary  that  the  relative  slip  from 
synchronism  be  constant,  which  condition  can  conveniently 
be  obtained  by  driving  the  rotor  with  a  synchronous  motor 
operated  from  the  same  supply  system  as  the  primary. 

Evidently  the  synchronous  motor  will  demand  power  from 
the  supply  in  addition  to  that  demanded  by  the  primary  circuit 
of  the  frequency  converter.  The  amount  of  this  power  is  de- 
termined by  the  speed  of  the  synchronous  motor  and  the  torque 
exerted  by  the  frequency  converter.  When  the  slip  of  the 
converter  is  unity,  the  power  demanded  by  the  synchronous 
motor  is  zero;  when,  however,  the  slip  is' two — that  is,  when 
the  frequency  of  the  secondary  is  twice  that  of  the  primary— 
the  power  demanded  by  the  synchronous  motor  is  equal  to 
that  demanded  by  the  primary  of  the  converter.  A  further 


INDUCTION  MOTORS  AS  FREQUENCY  CONVERTERS.     33 

analysis  will  show  that  the  power  supplied  by  the  frequency 
converter  bears  to  the  total  output  the  ratio  of  the  primary 
frequency  to  that  of  the  secondary,  the  remaining  power  being 
supplied  by  the  synchronous  motor. 

CAPACITY  OF  FREQUENCY  CONVERTERS. 

Since  the  total  output  appears  at  the  secondary  of  the  con- 
verter it  behooves  us  to  ascertain  in  what  manner  the  power 
supplied  by  the  synchronous  motor  enters  the  converter  wind- 
ings. 

Let  the  ratio  of  primary  to  secondary  turns  be  unity,  and  let 
us  consider  the  secondary  current  in  phase  with  the  secondary 
e.m.f.,  and  let  it  be  counter-balanced  by  an  equal  current  in 
the  primary  in  phase  with  its  e.m.f.,  then 

/  =  primary  current  in  phase  with  the  primary  e.m.f. 
and  in  phase  opposition  to  the  secondary  current. 

/          =  secondary  current. 

E         =  primary  impressed  e.m.f. 

S  E  =  secondary  generated  e.m.f.,  where  5  =  slip  with 
synchronism  as  unity. 

IE      =  primary  power. 

I  S  E  =  secondary  electrical  power. 

In  the  ordinary  induction  motor,  where  5  is  less  than  unity, 
the  secondary  generated  power  (S  I  E)  is  dissipated  in  the 
copper  of  the  secondary  windings,  while  the  remaining  power 
received  from  the  primary  (/  E  —  S  I  E)  is  available  for  mechan- 
ical work  When  5  £  is  equal  to  unity  the  secondary  generated 
power  (7  E)  is  totally  available  as  electrical  power,  which  may 
be  bst  in  the  resistance  of  the  secondary  windings  or  usefully 
applied  to  external  work,  while  the  mechanical  power  of  the 
secondary  is  zero.  When  S  is  greater  than  unity  the  total 
secondary  generated  power  (S  7  E)  is  available  at  the  secondary 
terminals.  Of  this  power  (7  E)  is  supplied  by  the  primary, 
while  the  remainder  is  supplied  by  the  synchronous  motor. 

It  is  seen,  therefore,  that  the  effect  of  driving  the  secondary 
backwards  is  to  increase  the  secondary  pressure  above  that  of 
the  primary,  and  that  the  power  for  such  increase  is  derived 
from  the  synchronous  motor. 

A  moment's  reflection  will  show  that  there  is  only  partial 
double  transformation  of  power  with  a  frequency  converter, 


34  ALTERNATING  CURRENT  MOTORS. 

and  that  the  sum  of  the  capacities  of  the  synchronous  motor 
and  the  converter  must  just  equal  the  output  plus  the  inevitable 
losses  in  each  machine.  A  numerical  example  will  show  this 
quite  plainly.  If  we  assume  a  lighting  load  of  60  kilowatts  to 
be  changed  in  frequency  from  25  to  60  cycles,  then  the  capacity 
of  the  frequency  converter  proper  must  be  25  kilowatts,  and 
that  of  the  synchronous  motor  35  kilowatts,  as  shown  above. 
It  should  be  noted,  however,  that  while  the  iron  loss  of  the  con- 
verter primary  is  the  same  as  that  of  a  25-kw.  induction  motor 
on  25  cycles,  the  iron  loss  of  the  secondary  of  the  converter  is 
that  of  a  60-cycle,  60-kw.  generator,  and  thus  very  materially 
greater  than  that  of  a  25-kw.  induction  motor,  which  latter, 
in  fact,  is  usually  quite  negligible. 

By  over-excitation,  the  leading  component  of  the  current 
demanded  by  the  synchronous  motor  may  be  adjusted  to  equal 
the  lagging  component  due  to  the  exciting  current  of  the  fre- 
quency converter,  so  that  the  external  apparent  power  factor 
of  the  equipment  may  be  kept  quite  high. 


CHAPTER  V. 
THE   HEYLAND   INDUCTION   MOTOR. 

EXCITATION  OF  INDUCTION  MOTORS. 

When  there  is  impressed  upon  the  primary  of  an  induction 
motor  an  e.m.f.  of  full  normal  value,  the  secondary  being  on 
open  circuit  and  the  rotor  stationary,  there  is  caused  to  flow 
within  the  windings  a  current  the  magnitude  and  phase  of 
which  can  best  be  determined  by  ascertaining  the  value  of  its 
two  components.  Its  power  component  is  of  a  value  such  that 
its  product  with  the  circuit  e.m.f.  is  just  sufficient  to  supply 
to  the  motor  the  no-load  power,  consisting  of  primary  copper 
loss  and  both  primary  and  secondary  iron  loss.  Its  wattless 
or  quadrature  component  has  a  value  such  that  its  product 
with  the  number  of  turns  of  the  primary  windings  produces 
the  requisite  magnetomotive  force  to  cause  to  flow,  through 
the  reluctance  of  its  paths,  a  magnetic  flux  of  which  the 
rate  of  change  at  each  instant  is  proportional  to  the  otherwise 
unbalanced  primary  e.m.f.  The  unbalanced  primary  e.m.f. 
referred  to  is  that  portion  of  the  impressed  e.m.f.  which  re- 
mains after  subtracting  (in  proper  phase  position)  the  e.m.f. 
consumed  by  the  local  impedance  of  the  primary  windings 
due  to  the  passage  of  the  current  therein.  The  observed  pri- 
mary current  is  the  vector  sum  of  the  above  named  components, 
while  the  phase  relation  of  the  current  with  respect  to  the 
e.m.f.  is  given  by  that  angle  the  cosine  of  which  is  equal  to  the 
power  component  divided  by  the  observed  current. 

It  is  evident  that  in  order  to  decrease  the  phase  displacement 
between  the  current  and  e.m.f.,  it  is  necessary  to  render  the 
power  component  and  the  observed  current  more  nearly  equal 
in  value.  This  may  be  accomplished  by  one,  or  both,  of  two 
methods;  i.e.,  increasing  the  power  component,  or  decreasing 
the  wattless  component  of  the  current.  The  former  method 
is  that  to  which  is  due  the  increased  power  factor,  or  decreased 
phase  displacement,  with  increase  of  load  on  the  ordinary  in- 

35 


36  ALTERNATING  CURRENT  MOTORS. 

duction  motor,  the  maximum  power  factor  appearing  when  that 
condition  is  reached  when  the  ratio  of  the  power  component  to 
the  wattless  component  is  a  maximum.  If  the  wattless  com- 
ponent remained  constant  in  value  an  increase  in  load  would 
always  be  accompanied  by  an  increase  in  power  factor.  But 
owing  to  the  existence,  upon  increase  of  primary  and  secondary 
currents,  of  local  magnetic  lines  of  force  which  flow  in  closed 
paths  around  the  primary  coil  without  enclosing  any  secondary 
conductors  and  around  the  secondary  without  enclosing  any 
primary  conductors,  producing  field  distortion  and  a  reduction 
of  the  flux  common  to  both  windings,  the  wattless  component 
is  increased  with  each  increment  of  load  and  the  rate  of  increase 
is  much  augmented  as  full  load  is  approached  and  passed.  Ob- 
viously, therefore,  the  result  of  loading  an  induction  motor  is 
not  to  decrease  the  lagging  current  below  that  value  which  the 
motor  demands  at  no-load,  though  the  increased  power  factor 
is  frequently  credited  with  reducing  the  undesirable  effects 
which  are  caused  by  and  proportional  to  the  lagging  wattless 
component  of  the  primary  current. 

The  most  logical  method  of  improving  the  power  factor  of 
operation  of  an  induction  motor  is  that  which  has  for  its  object 
the  reduction  of  the  wattless  component  of  the  primary  current. 
The  employment  of  details  of  construction  which  lessen  the 
tendency  of  the  flux  to  close  around  one  winding  without  en- 
circling the  other,  tends  to  the  reduction  of  the  wattless  current 
component  with  increase  of  load  and  thus  improves  the  operating 
power  factor  and  the  mechanical  performance  of  the  motor, 
since  the  maximum  torque  which  the  rotor  can  exert  and  the 
overload  which  the  motor  can  carry  are  determined  almost 
wholly  by  the  local  reactance  of  the  windings. 

The  amount  of  magnetic  flux  which  the  no-load  wattless 
component  of  the  current  must  produce  is  determined  by  the 
primary  e.m.f.  and  frequency,  and  the  value  of  the  wattless 
component  depends  upon  the  reluctance  of  the  paths  which 
these  lines  must  take.  The  major  portion  of  this  reluctance 
is  found  in  the  air-gap  which  forms  the  separating  space  be- 
tween the  rotor  and  stator.  A  reduction  in  the  air-gap  is 
followed  by  an  almost  proportional  decrease  in  the  inductive 
or  wattless  component  of  the  current,  which  accounts  for  the 
fact  that  all  modern  induction  motors  are  constructed  with 


THE  HEYLAND  INDUCTION  MOTOR.  37 

air-gaps  as  small  as  permitted  by  the  requisite  mechanical 
clearance  between  the  stationary  and  revolving  members  of 
the  machine.  It  is  evident  that  much  further  improvement 
along  this  line  can  scarcely  be  expected. 

Although  it  is  for  the  purpose  of  producing  counter  e.m.f. 
in  the  primary  that  the  core  magnetism  is  required,  it  is  not 
essential  that  the  exciting  current  should  flow  in  the  primary 
windings;  the  magnetomotive  force  for  excitation  may  with 
equal  effect  be  ^upplied  by  current  in  the  secondary,  although 
the  ampere-turns  in  the  one  case  must  equal  those  in  the  other. 

SECONDARY  EXCITING  MAGNETOMOTIVE  FORCE. 

Consider  an  induction  motor  with  a  stationary  primary  and 
a  revolving  secondary,  the  latter  being  coil  wound,  and,  for 
simplicity  of  discussion,  assume  the  secondary  turns  equal  in 
number  to  the  primary.  Let  the  rotor  be  at  rest  and  the 
secondary  on  open  circuit.  On  applying  normal  e.m.f.  to  the 
primary  windings  there  will  flow  a  current  having  power  and 
wattless  components,  as  discussed  previously.  If  the  secondary 
be  now  interchanged  for  the  primary  winding  in  its  connection 
to  the  external  circuit,  an  exactly  similar  and  equal  current  will  be 
found  to  flow  in  the  secondary  winding.  If  both  the  primary  and 
secondary  windings  be  connected  simultaneously  to  the  ex- 
ternal supply  circuits,  care  being  taken  that  the  separate  re- 
volving fields  due  to  the  wattless  current  components  in  the 
two  windings  travel  around  the  air-gap  in  the  same  direction, 
it  will  be  found  that  at  a  certain  position  of  the  rotor  relative 
to  the  stator  the  two  windings  combined  take  a  current  equal 
in 'value  and  character  to  that  taken  by  each  winding  when 
connected  up  alone.  By  increasing  slightly  the  e.m.f.  im- 
pressed upon  the  secondary,  the  primary  e.m.f.  remaining  as 
before,  the  wattless  component  flowing  in  the  primary  windings 
may  be  reduced  to  zero,  or  even  given  a  negative  value.  Under 
this  condition  the  quadrature  component  of  the  ampere-turns 
in  the  secondary  windings  has  a  value  just  sufficient  to  give 
the  magnetomotive  force  to  produce  the  requisite  core  magnetism 
to  induce  the  desired  primary  counter  e.m.f.  The  rotor  is, 
however,  stationary,  and  even  resists  any  attempt  at  motion 
in  either  direction  so  that  with  connections  arranged  as  just 
described  no  mechanical  power  can  be  obtained  from  the  motor. 


38  ALTERNATING  CURRENT  MOTORS. 

It  is  possible  to  supply  the  exciting  magnetomotive  force  to 
the  motor  through  the  secondary  windings  while  the  rotor 
travels  at  full  speed  without  any  constructive  change  in  the 
windings  of  the  motor,  as  here  assumed.  Consider  normal 
e.m.f.  impressed  upon  the  primary  with  the  secondary  on  closed 
circuit  and  the  rotor  traveling  at  full  speed.  The  wattless  com- 
ponent of  the  primary  current  will  have  practically  the  same 
value  as  with  the  secondary  on  open  circuit  and  the  rotor  sta- 
tionary. If  now  there  be  introduced  into  the  secondary  windings 
direct  current  adjusted  in  value  to  equal  the  mean  effective 
value  of  the  wattless  component  of  the  primary  current,  as 
previously  observed,  it  will  be  found  that  the  lagging  com- 
ponent of  the  primary  current  has  been  reduced  to  zero,  so  that 
the  power  factor  has  increased  to  unity.  The  motor  may  now 
be  given  its  full  load  and  its  performance  will  be  found  to  be 
satisfactory  in  all  respects. 

DIRECT  CURRENT  IN  SECONDARY  COILS. 

It  is  well  at  this  point  to  investigate  the  advantages  and  dis- 
advantages of  this  latter  arrangement.  The  core  magnetism 
is  not  decreased  in  value,  so  that  the  exciting  current  is  as  large 
as  ever,  but  the  latter  has  been  transferred  from  the  primary 
to  the  secondary  winding,  and  not  only  is  its  value  the  same 
as  before,  but  because  of  the  fact  that  when  existing  in  the 
primary  windings  it  was  the  wattless  component  of  the  working 
current,  its  heating  effect  upon  the  secondary  windings  is  just 
equal  to  the  reduction  in  heat  loss  in  the  primary  windings; 
therefore  no  gain  has  been  made  through  a  reduction  in  heat 
loss. 

Let  E  =  primary  e.m.f. 

R   =  primary  or  secondary  resistance. 

IP  =  power  component  of  primary  current. 

lq  =  exciting  current,   whether  in  the  primary   or  sec- 

ondary windings. 

Under  normal  no-load  conditions,  the  primary  copper  loss  is 


+  I^R  -V 

while  the  secondary  copper  loss  at  no-load  is  of  practically  zero 
value. 


THE  HEY  LAND  INDUCTION  MOTOR.  39 

With  direct  current  in  the  secondary  winding,  the  copper  loss 
there  is 

!£R, 

and  the  primary  copper  loss  is 


so  that  the  total  copper  loss,  I  p-  R  +  IJ  R,  in  the  two  windings 
is  the  same  as  before. 

Although  the  actual  excitation  loss  in  the  primary  under 
normal  no-load  cond  tions  was  Iq~  R,  the  apparent  exciting 
power  was  /  q  E  in  addition  to  the  loss.  This  "wattless" 
current,  /  q,  is  required  to  travel  over  the  circuits  and  trans- 
forming devices  between  the  generator  and  motor,  and  tends 
to  decrease  the  field  magnetism  of  the  generator  and  to  reduce 
further  the  e.m.f.  at  the  receiving  end  of  the  transmission  line. 
An  elimination  of  this  wattless  component  removes  the  greatest 
objection  to  the  otherwise  admirable  induction  motor. 

With  direct  current  supplied  to  its  secondary  windings  an 
induction  motor  travels  at  synchronous  speed,  and,  in  fact,  be- 
comes transformed  to  a  synchronous  motor  and  therefore  pos- 
sesses all  of  the  qualities  inherent  in  the  performance  of  this 
type  of  machine.  Mr.  Heyland  has  devised  a  method,  however, 
by  which  approximately  unidirectional  current  may  be  supplied 
to  the  secondary  windings  while  the  motor  yet  retains  the 
characteristics  of  the  asynchronous  type  of  machine. 

ALTERNATING  CURRENT  IN  SECONDARY  COILS. 

The  method  of  obtaining  this  most  desirable  result  consists 
in  applying  to  the  secondary  windings  a  commutator  to  which 
current  from  the  source  of  supply  for  the  primary  is  led  by 
way  of  properly  disposed  brushes.  The  commutator  necessary 
for  the  operation  of  the  motor  is  small  in  size,  and,  on  account 
of  the  fact  that  adjacent  segments,  are  connected  together  by 
non-inductive  resistance  external  to  the  windings,  there  is  no 
possibility  of  sparking  at  the  brushes,  and  its  performance  is 
quite  similar  to  that  of  slip  rings. 

Fig.  18  represents  diagrammatically  a  direct-current  armature 
complete  with  commutator,  to  be  used  as  the  secondary  of  an 
induction  motor.  Since  no  current  whatever  will  flow  in  the 
conductors  of  a  direct  current  armature  when  on  open  circuit, 


40  ALTERNATING  CURRENT  MOTORS. 

the  armature  alone  will  possess  no  tendency  to  be  drawn  into 
rotation  by  the  revolving  magnetism  of  the  primary.  In 
order  that  the  armature  winding  may  serve  as  the  secondary 
of  the  induction  motor,  it  is  necessary  that  points  on  the  ar- 
mature possessing  difference  of  potential  be  joined  together. 
The  resistance,  shown  in  Fig.  18  as  being  connected  between 
adjacent  segments,  serves  to  complete  the  secondary  circuit, 
and  with  the  brushes  removed  from  the  commutator  the  motor 
thus  equipped  will  operate  in  all  respects  similarly  to  one  with 
a  pure  "  squirrel-cage  "  secondary  winding.  The  three  brushes 
shown  in  the  figure  are  for  the  purpose  of  allowing  the  intro- 


FIG.  18. — Direct-current  Armature  of  Heyland  Motor  for 
Three-phase  Secondary  Excitation,  Showing  Segment- 
connecting  Resistances. 

duction  of  three-phase  current  into  the  secondary  for  supplying 
sufficient  magnetomotive  force  for  field  excitation,  in  order 
that  no  wattless  current  need  flow  in  the  primary  windings. 

According  to  the  foregoing  discussions  it  is  plain  that  with 
the  rotor  traveling  at  synchronous  speed,  current  of  zero  fre- 
quency may  be  utilized  to  supply  the  magnetomotive  force  for 
excitation,  while  with  the  rotor  stationary,  current  at  a  fre- 
quency equal  to  that  of  the  primary  may  thus  be  employed.  A 
little  further  consideration  will  convince  one  that  at  speeds 
between  synchronism  and  standstill  current  at  intermediate 
frequencies  may  be  so  used,  and  that  the  requisite  frequency 
in  each  case  is  equal  to  the  product  of  the  percentage  of  slip 


THE  HEY  LAND  INDUCTION  MOTOR.  41 

and  the  primary  frequency.  By  attaching  to  the  secondary 
windings  a  commutator,  to  the  brushes  upon  which  current  is 
led  at  the  primary  frequency,  the  current  in  the  secondary  will, 
at  any  speed,  possess  the  frequency  required  for  excitation. 

ACTION  WITH  STATIONARY  ROTOR. 

For  the  sake  of  simplicity  in  explanation,  consider,  in  the  first 
place,  that  upon  the  rotor  there  is  placed  a  symmetrical  direct- 
current  armature  winding  with  a  corresponding  commutator. 
By  introducing  into  the  windings  an  alternating  current  at  the 

TABLE  I. — Instantaneous  values  of  currents  in  each  coil  of    direct-current    winding; 
three-phase  excitation;  rotor  stationary.     (Fig.  18). 


Number  of 
coil. 

A 

B 

C 

D 

E 

F 

1 

+  10.00 

+  8.66 

+  5.00 

0.00 

—  5.00 

—8.66 

2 

+  10.00 

+  8.66 

+  5.00 

0.00 

—  5.00 

—8.66 

3 

+  10.00 

+  8.66 

+  5.00 

0.00 

—  5.00 

—8.66 

4 

+  10.00 

+  8.66 

+  5.00 

0.00 

—  5.00 

—8.66 

5 

—  5.00 

—8.66 

—10.00 

—8.66 

—  5.00 

0.00 

6 

—  5.00  . 

—8.66 

—10.00 

—8.66 

—  5.00 

0.00 

7 

—  5.00 

—8.66 

—10.00 

—8.66 

—  5.00 

0.00 

8 

—  5.00 

—8.66 

—10.00 

—8.66 

—  5.00 

0.00 

9 

—  5.00 

0.00 

+  5.00 

+  8.66 

+  10.00 

+  8.66 

10 

—  5.00 

0.00 

+  5.00 

+  8.66 

+  10.00 

+  8.66 

11 

—  5.00 

0.00 

+  5.00 

+  8.66 

+  10.00 

+  8.66 

12 

i 

-  5.00 

0.00 

+  5.00 

+  8.66 

+  10.00 

+  8.66 

primary  frequency,  when  the  rotor  is  stationary,  the  effect  will 
be  exactly  the  same  as  was  previously  found  when  the  current 
at  the  same  frequency  was  supplied  to  the  secondary  of  the 
ordinary  induction  motor  with  stationary  rotor;  that  is,  by 
adjustment  of  secondary  current,  the  wattless  component  of 
the  primary  current  may  be  made  to  disappear.  If,  however, 
the  rotor  be  given  a  certain  speed,  the  same  current  in  the  sec- 
ondary will  continue  to  produce  the  same  magnetizing  effect  as 
at  standstill,  for  at  each  instant  the  commutator  will  cause 
the  current  to  traverse  conductors  occupying  the  same  position 


42 


ALTERNATING  CURRENT  MOTORS. 


in  space,  and  the  effect  of  the  secondary  current  upon  the  pri- 
mary core  magnetism  will  not  be  altered. 


FIG.  19. — Currents  for      FIG.  20. — Currents  for      FIG.  21.— Currents  in 
Stationary  Armature.         Armature  at  Syn-  Double-wound  Arma- 

chronous  Speed.  ture. 

Fig.  19  represents  such  symmetrical  armature  winding  upon 
the  commutator  of  which  are  placed  three  brushes  for  the  in- 
troduction of  three-phase  current  for  excitation.  The  in- 


THE  HEY  LAND  INDUCTION  MOTOR.  43 

stantaneous  values  of  the  current  in  the  individual  coils  as  the 
current  in  the  three-phase  leads  changes  value  are  indicated 
for  each  30  degrees  increment  of  time  in  the  several  diagrams 
of  Fig.  19,  and  collectively  recorded  in  Table  I.  The  rotor  is 
here  assumed  to  be  stationary.  It  will  be  observed  that  through- 
out each  cycle  the  current  in  each  coil  undergoes  a  double 
reversal  and  reaches  full  normal  value  in  both  positive  and  nega- 
tive directions.  The  reactive  e.m.f.  induced  in  the  windings,  which 
is  proportional  to  the  rate  of  change  of  the  local  flux  surrounding 
the  conductor  due  to  the  current  flowing  therein,  is  therefore 

TABLE    II. — Instantaneous   values  of  currents  in   each   coil  of  direct-current   winding; 
three-phase  excitation;  synchronous  speed.     (Fig.   19.) 


Number  of 
coil. 

A 

B 

C 

D 

E 

F 

, 

+  10.00 

0.00 

+  5.00 

+  8.66 

+  10.00 

0  00 

2 

+  10.00 

+  8.66 

+  5.00 

+  8.66 

+  10  00 

+  8.66 

3 

+  10.00 

+  8.66 

+  5.00 

+  8.66 

+  10.00 

+  8.66 

4 

+  10.00 

+  8.66 

+  5.00 

0.00 

+  10.00 

+  8.66 

5 

-  o.OO 

+  8.66 

+  5  00 

0.00 

-  5.00 

+  8.66 

6 

-  5.00 

-8.66 

+  5.00 

0.00 

-  5.00 

—8.66 

7 

-  5.00 

—8.66 

—  10.00 

0.00 

-  5.00 

—8.66 

8 

-  5.00 

—8.66 

—10.00 

—8.66 

-  5.00 

—8.66 

9 

-  5.00 

—8.66 

—  10.00 

—8  66 

-  5.00 

—8.66 

10 

-  5.00 

0  00 

—10.00 

-8.66 

—  5.00 

0.00 

11 

—  5.00 

0.00 

+  5.00 

—8.66 

-  5.00 

0.00 

12 

-  5.00 

0  00 

+  5.00 

+  8.66 

-  5.00 

0.00 

of  a  value  corresponding  to  the  primary  frequency,  and  there 
is  required  for  the  excitation  a  wattless  component  of  e.m.f. 
equal  to  that  which  would  have  been  required  had  the  exciting 
current  been  allowed  to  flow  in  the  primary  windings. 

ACTION  WITH  ROTOR  AT  SYNCHRONOUS  SPEED. 

When  the  rotor  is  traveling  at  a  speed  approximately  syn- 
chronism the  exciting  current  required  in  the  secondary  windings 
is  of  the  same  value  as  before,  but  the  requisite  wattless  com- 
ponent is  much  reduced  because  of  the  decrease  in  the  reactive 


44  ALTERNATING  CURRENT  MOTORS. 

e.m.f.  of  the  secondary  windings.  With  an  infinite  number  of 
commutator  segments  and  of  phases  for  the  exciting  current 
in  the  secondary,  the  reactive  e.m.f.  would  entirely  disappear 
at  synchronous  speed,  and  it  would  increase  directly  with  the 
rotor  slip.  Fig.  20  indicates  the  changes  in  the  value  of  the 
secondary  current  in  the  individual  coils  with  three-phase 
excitation,  when  the  rotor  is  traveling  at  synchronous  speed, 
For  the  sake  of  clearness  the  windings  are  considered  stationary 
and  the  brushes  are  supposed  to  revolve  at  synchronous  speed, 
the  effect  being  the  same  as  with  stationary  brushes  and  re- 
volving windings,  of  course. 

A  glance  at  Table  II  will  show  that  the  fluctuations  in  the  cur- 
rent in  the  individual  coils  are  much  reduced  at  synchronous 
speed,  and  consequently  the  flux  around  the  conductors,  to  the 
rate  of  change  of  which  is  due  the  reactive  e.m.f.,  has  a  much 
more  nearly  constant  value  than  when  the  secondary  is  sta- 
tionary, and  thus  the  necessary  wattless  component  for  second- 
ary excitation  is  correspondingly  diminished. 

The  value  of  the  e.m.f.  for  excitation  will  depend  upon  the 
number  of  and  resistance  of  the  secondary  conductors  and  upon 
the  reactive  e.m.f.,  and  will  in  general  be  much  below 
that  required  for  the  primary  windings.  It  can  con- 
veniently be  obtained  by  transformation  from  the  supply 
circuit.  An  increase  in  the  excitation  e.m.f.  above  normal 
value  will  cause  the  primary  to  draw  leading  currents,  the  value 
of  which  may  be  adjusted  to  equal  the  lagging  current  de- 
manded by  the  primary  of  the  excitation  transformers,  so  that 
the  motor  and  transformers  considered  as  a  unit  may  be  oper- 
ated at  unity  power  factor. 

FUNCTION  OF  THE  CONNECTING  RESISTANCES. 

In  order  that  the  rotor  shall  be  drawn  into  rotation  by  the 
revolving  field,  it  is  necessary  that  the  secondary  circuit  be  closed. 
This  is  accomplished  by  connecting  resistance  between  the 
adjacent  commutator  segments,  as  indicated  at  position  E  of 
Fig.  20.  Since  the  current  induced  in  the  secondary,  due  to 
the  slip  of  the  conductors  relative  to  the  revolving  field,  must 
pass  through  these  resistance  coils,  which  are,  in  fact,  in  series 
with  the  secondary  windings  of  the  machine  considered  as  an 
induction  motor,  it  is  desirable  to  make  the  resistance  as  low 


THE  HEYLAND  INDUCTION  MOTOR.  45 

as  possible,  for  the  slip  due  to  any  given  torque  and  the  sec- 
ondary loss  due  to  a  given  current  depend  directly  on  the  total 
resistance  of  the  secondary  circuit.  But  on  account  of  the  fact 
that  these  resistances  form  a  parallel  path  for  the  exciting 
current  from  the  lowering  transformers,  it  is  desirable,  in  order 
to  decrease  the  shunt  loss,  that  the  resistance  be  large.  The 
resistance  is,  therefore,  given  a  value  large  in  comparison  with 

TABLE  III. — Instantaneous  values  of  currents  in  each  slot;  three-phase  winding;  three- 
phase  excitation;  synchronous  speed.     (Fig.  20.) 


CO 

11 

o  rt 
0^. 

is 

A 

B 

C 

D 

E                  F 

v 

17.32 

7.50 

0.00 

7.50 

17.32               7.50 

1" 

0.00 

7.50 

17.32 

7.50 

0.00               7.50. 

2' 

0.00 

7.50 

8.66 

0.00 

0.00               7.50 

2" 

8.66 

0.00 

0.00 

7.50 

8.66               0.00 

3' 

8.66 

7.50 

0.00 

0.00 

8.66               7.50 

3" 

0.00 

0.00 

8.66 

7.50 

0.00               0.00 

Coils  in  slots 
1.2  and  3. 

1 

17.32 

15.00 

17.32 

15.00 

17.32             15.00 

2 

8.66 

7.50 

8.66 

7.50 

8.66               7.50 

3 

8.66 

7.50 

8.66 

7.50 

8.66               7.50 

that  of  the  armature  winding,  but  small  enough  not  to  increase 
unduly  the  total  secondary  resistance. 

The  resistance  forms  a  non-inductive  shunt  to  the  inductive 
windings  of  the  armature,  and  thus  serves  the  very  advantageous 
purpose  of  suppressing  any  sparking  at  the  excitation  brushes, 
which  fact  allows  the  use  of  copper  brushes  of  cross-section 
much  less  than  that  required  for  carbon  brushes  and  the  com- 
mutator is  given  a  correspondingly  smaller  size.  So  effective 
is  the  elimination  of  sparking  that  no  observable  result  is  pro- 
duced by  shifting  the  brushes  into  any  position. 


46  ALTERNATING  CURRENT  MOTORS. 

DOUBLE  THREE-PHASE  SECONDARY  WINDING. 

In  order  to  reduce  the  reactive  e.rn.f.  of  the  secondary  below 
that  value  which  it  is  possible  to  obtain  with  the  direct-current 
winding,  there  has  been  developed  a  double,  semi-parallel  con- 
nected three-phase  winding  which  possesses  many  advantages. 
This  winding  is  diagrammatically  represented  in  Fig.  21,  which 
indicates  also  the  changes  in  value  of  the  secondary  current 
when  the  rotor  (brush)  is  traveling  at  synchronous  speed. 
The  winding  is  connected  double  star  fashion.  Coils  I'  and  1" 
lie  in  the  same  slots;  coils  2'  and  2"  lie  in  the  same  slots,  and 
the  same  is  true  of  coils  3'  and  3".  The  object  of  this  arrange- 
ment is  to  maintain  as  nearly  constant  as  possible  the  local 
flux  surrounding  each  individual  conductor,  and  thereby  reduce 
as  far  as  possible  the  reactive  e.m.f.  The  flux  encircling  a 
slot  depends  upon  the  current  flowing  through  the  slot,  but 
it  is  not  altered  by  a  change  of  the  current  from  one  conductor 
to  another  in  the  same  slot.  By  the  present  arrangement, 
when  the  current  in  one  conductor  occupying  a  certain  slot 
has  its  zero  value  the  other  conductor  carries  its  maximum 
current,  and,  as  Table  III  shows,  the  actual  current  which  causes 
local  flux  around  the  separate  coils  undergoes  but  slight  varia- 
tion in  value.  The  reactive  e.m.f.  is,  therefore,  quite  small, 
and  the  secondary  wattless  component  is  much  reduced,  so 
that  the  exciting  transformers  may  be  of  small  capacity  and 
operated  at  high-power  factor;  moreover,  a  relatively  slight 
increase  in  e.m.f.  at  the  excitation  brushes  causes  the  leading 
current  in  the  primary  windings  of  the  motor  to  compensate 
for  the  lagging  current  in  the  transformers,  giving  to  the  ma- 
chine, as  a  whole,  a  power  factor  of  unity.  By  adjusting  the 
degree  of  over-excitation  at  no-load  and  shifting  the  secondary 
brushes  the  requisite  amount  in  the  proper  direction,  the  com- 
pensation may  be  made  such  as  to  hold  the  power  factor  at 
practically  unity  at  all  loads  within  the  operating  range  of  the 
motor. 

The  three-phase  winding  allows  of  the  introduction  of  resist- 
ance in  the  secondary  windings  by  means  of  slip  rings  connected 
at  the  center  of  the  "  star,"  and  the  starting  torque  or  operating 
speed  can  thus  conveniently  be  varied  at  will. 

On  account  of  the  reduction  in  the  effective  secondary  react- 
ance and  the  elimination  of  the  wattless  current  component  in 


THE  HEY  LAND  INDUCTION  MOTOR.  47 

the  primary  windings,  the  compensated  induction  motor  may 
be  given  a  much  greater  load  than  the  corresponding  motor 
without  compensation,  while  the  ability  to  supply  even  an 
enormously  increased  magnetomotive-  force  for  excitation 
allows  the  compensated  motor  to  be  given  a  larger  ratio  of 
current-conductor  material  to  magnetic  material,  especially 
upon  the  secondary,  in  which  the  iron  loss  in  any  case  is  rela- 
tively small.  Consequently,  by  the  use  of  the  specially-wound 
secondary  with  commutator  and  transformers,  the  capacity  of 
a  given  induction  motor  may  be  much  increased,  or,  and  for  the 
same  reasons,  an  induction  motor  of  a  given  capacity  may  thus 
be  constructed  at  a  proportionately  less  cost. 
-  It  is  scarcely  probable  that  the  mere  elimination  of  the  watt- 
less current  component  from  the  circuit  wires  would  prove 
sufficient  inducement  to  a  consumer  to  justify  the  extra  ex- 
pense of  adding  a  commutator  and  the  accompanying  complica- 
tions to  the  one  piece  of  reliable  machinery,  the  simplicity  of 
which  has  previously  been  the  characteristic  that  led  most  rap- 
idly to  its  adoption  in  preference  to  commutator  motors.  The 
ability  of  manufacturers  to  produce  at  a  reduced  cost  motors 
giving  satisfactory  service  to  the  purchaser  can  alone  cause  the 
compensated  motor  to  compete  successfully  with  its  highly 
efficient  and,  above  all,  simple  rival. 


CHAPTER  VI. 
THE  SINGLE-PHASE  INDUCTION  MOTOR. 

OUTLINE  OF  CHARACTERISTIC  FEATURES. 

While  under  no  condition  is  the  single-phase  motor  more 
satisfactory  or  economical  than  the  polyphase  machine,  yet, 
by  a  little  care  in  the  selection  of  a  motor  for  the  service  re- 
quired, the  performance  of  the  single-phase  machine  may 
compare  quite  favorably  with  that  of  the  polyphase  type. 
The  most  prominent  difference  between  the  single-phase  and 
the  polyphase  motor  is  the  inability  of  the  former  to  exert  a 
torque  at  standstill.  Numerous  devices  have  been  applied  .to 
render  single-phase  motors  self-starting,  which  have  met  with 
varying  success.  A  difficulty  which  the  designer  has  had  to 
encounter  lies  in  the  fact  that,  with  few  exceptions,  such  de- 
vices are  applicable  only  to  motors  of  small  sizes,  or  where 
efficiency  is  of  small  moment.  Little  trouble  is  experienced 
in  designing  self-starting  single-phase  motors  for  meter  or  fan 
work,  but  the  problem  assumes  a  different  aspect  when  motors 
for  power  purposes  are  desired. 

In  what  follows,  an  attempt  will  be  made  to  outline  the 
characteristic  features  of  the  single-phase  induction  motor,  to 
ascertain  the  similarities  and  the  differences  between  the  per- 
formance of  a  single-phase  and  that  of  a  polyphase  machine, 
and  to  investigate  the  methods  by  which  the  single-phase  motor 
may  be  operated  under  various  conditions.  The  graphical 
representation  of  the  phenomena  of  the  single-phase  motor  is 
reserved  for  a  subsequent  chapter. 

Although  supplied  with  current,  which,  if  acting  alone, 
could  produce  only  a  simple  alternating  magnetism,  in  contra- 
distinction to  a  rotating  field,  it  is  found  that  a  single-phase 
motor  under  operating  conditions  develops  a  rotating  field 
essentially  the  same  as  would  be  obtained  were  the  machine 
operated  on  a  polyphase  circuit.  The  effect  of  the  mechanical 

48 


THE  SINGLE-PHASE  INDUCTION  MOTOR. 


49 


motion  of  the  secondary  in  producing  the  rotating  field  may 
be  determined  as  follows: 

PRODUCTION  OF  QUADRATURE  MAGNETISM. 

Assume  for  the  purpose  of  illustration  a  motor  with  four 
mechanical  poles  having  the  two  opposite  poles  excited  by  a 
single-phase  alternating  current,  and  consider  the  moment 
when  one  of  these  poles  is  at  a  maximum  north  and  the  other 
a  maximum  south,  as  shown  in  Fig.  22.  If  the  rotor  be  moving 
across  this  field  in  the  direction  indicated,  there  will  be  gen- 
erated in  each  of  the  conductors  under  the  poles  an  e.m.f.  pro- 
portional to  the  product  of  the  field  magnetism  and  speed  of 


FIG.  22. — Production  of  Quadrature  Magnetism. 

rotor.  Evidently  if  the  speed  be  constant,  of  whatsoever 
value,  this  e.m.f.  will  vary  directly  with  the  strength  of  mag- 
netism; that  is,  it  will  be  maximum  when  the  magnetism  is 
maximum,  and  zero  at  zero  magnetism.  Other  conditions  re- 
maining the  same,  the  maximum  value  of  the  secondary  e.m.f. 
will  vary  directly  with  the  speed  of  the  rotor. 

If  the  circuits  of  the  rotor  conductors  be  closed,  there  will 
tend  to  flow  therein  currents  of  strengths  depending  directly 
upon  the  e.m.fs.  generated  in  the  conductors  at  that  instant 
and  inversely  upon  the  impedance  of  the  rotor  conductors.  The 
somewhat  unique  condition  of  e.m.fs.  in  a  combined  series  and 
parallel  circuit,  which  exists  in  the  rotor  as  here  described,  is 
depicted  by  analogy  in  Fig.  23,  where  the  e.m.f.  in  each  conductor 


50  ALTERNATING  CURRENT  MOTORS. 

across  the  rotor  core  is  represented  by  a  battery.  The  abso- 
lute value  of  each  e.m.f.  depends  upon  the  strength  of  the  pri- 
mary field  and  the  position  of  the  conductor  in  that  field,  and 
hence  changes  from  instant  to  instant.  The  current  which 
flows  through  the  end  rings  changes  its  direction  of  flow  with 
reference  to  the  field  poles  once  for  each  reversal  of  the  primary 
magnetism. 

The  current  which  flows  through  the  rotor  circuits  at  once 
produces  a  magnetic  flux  which,  by  its  rate  of  change  in  value 
generates  in  the  rotor  conductors  a  counter  e.m.f.,  opposing  the 
e.m.f.  that  causes  the  current  to  flow,  and  of  such  a  value  that 
the  difference  between  it  and  this  e.m.f.  is  just  sufficient  to 
cause  to  flow  through  the  impedance  of  the  conductors  a  current 
whose  magnetomotive  force  equals  that  necessary  to  drive  the 


FIG.  23. — Current  and  e.m.f.  in  Squirrel-cage  Secondary. 

required  lines  of  magnetism  through  the  reluctance  of  their 
paths.  Since  this  latter  magnetism  must  have  a  rate  of  change 
equal  (approximately)  to  the  e.m.f.  generated  in  the  rotor 
conductors  by  their  motion  across  the  primary  field,  and  since 
this  e.m.f.  is  in  time-phase  with  the  primary  field,  it  follows 
that  this  magnetism  must  have  a  value  proportional  to  the  rate 
of  change  of  the  primary  magnetism,  and,  if  the  primary  mag- 
netism follows  a  sine  curve  of  values,  this  magnetism  must 
follow  the  corresponding  cosine  curve;  that  is,  it  must  be  in 
quadrature  to  the  primary  magnetism  as  to  time  phase. 

Consider  the  N  pole  due  to  primary  magnetism  at  its 
maximum  strength,  and  decreasing  in  value.  The  e.m.f.  gen- 
erated in  the  rotor  will  tend  to  send  lines  of  force  at  right  angles 
to  the  primary  field,  inducing  a  secondary  N  pole  at  the  right 
(see  Fig.  22).  The  lines  of  secondary  magnetism  (induced  field) 


THE  SINGLE-PHASE  INDUCTION  MOTOR.  51 

continue  to  increase  in  number  so  long  as  the  primary  mag- 
netism does  not  change  direction  of  flow.  They,  therefore, 
reach  their  maximum  value  when  the  primary  magnetism  dies 
down  to  zero,  at  which  instant  the  induced  or  secondary  mag- 
netism will  have  its  maximum  strength  with  the  north  pole 
to  the  right,  as  drawn. 

As  the  primary  magnetism  now  shifts  its  north  pole  to  the 
bottom,  the  secondary  lines  begin  to  decrease  in  number,  and 
they  will  reach  their  zero  value  when  the  primary  magnetism 
reaches  its  maximum  strength.  The  induced  magnetism  will 
then  begin  to  increase  its  lines  in  the  reverse  direction,  pro- 
ducing a  north  pole  to  the  left,  and  it  will  reach  its  maximum 
strength  when  the  primary  magnetism  dies  down  to  zero  again. 
When  the  primary  magnetism  shifts  its  north  pole  back  to  the 
top,  the  secondary  magnetism  will  begin  to  decrease,  then  fi- 
nally build  up  with  its  north  pole  to  the  right  again,  and  so  on. 

The  north  magnetic  poles  produced  on  the  motor  thus  reach 
their  maximum  in  the  following  order:  top,  right,  bottom,  left, 
etc.,  or  in  the  direction  of  rotation.  Further  consideration  will 
show  that,  had  the  rotation  been  taken  in  the  opposite  direc- 
tion, the  poles  would  have  traveled  in  the  opposite  direction 
also. 

It  must  be  remembered  that  the  simultaneous  existence  of 
magnetic  fluxes  at  right  angles  in  the  same  material  is  entirely 
imaginary.  The  effect  of  each  is,  however,  real  and  the  ex- 
istence of  the  resultant  is  real. 

PRODUCTION  OF  REVOLVING  FIELD. 

When  the  rotor  is  traveling  at  synchronous  speed,  the  e.m.f. 
generated  in  the  secondary  conductors  by  their  motion  across 
the  primary  magnetism  is  of  such  a  value  as  to  require  the 
induced  field  (quadrature  magnetism)  to  be  equal  in  effective 
value  to  the  primary  field.  If  the  maximum  value  of  the  pri- 
mary field  be  taken  as  unity  and  time  be  denoted  in  angular 
degrees,  then  the  instantaneous  value  of  the  primary  magnetism 
may  be  represented  by  cos.  a,  where  a  measures  the  angle  of 
time  from  the  instant  of  maximum  primary  magnetism.  In 
a  similar  manner  sin.  a  may  represent  the  instantaneous  value 
of  the  quadrature  magnetism. 

These   two  fields  are  located  mechanically   90  degrees  from 


52 


ALTERNATING  CURRENT  MOTORS. 


each  other.  Fig.  24  indicates  the  manner  in  which  the  fluxes 
of  the  two  fields  vary  from  instant  to  instant,  and  the  position 
of  the  resultant  core  magnetism  at  each  instant,  a  two-pole 
motor,  with  a  ring-shaped  core  without  projecting  poles,  being 
assumed. 

O  B  =  O  C  cos.  a  =  value  and  position  of  primary  magnetism 

after  the  time  lapse  a. 

O  D  sin.  a  =  value  and  position  of    induced  magnetism 
at  same  instant. 


O  A 


OC  =  \/OB2xO  A2  represents  the  resultant  field,  both  in 
value  and  position.  The  point  P  describes  a  circle.  Without 
further  proof  it  is  evident  that,  neglecting  the  effect  of  local 


FIG.  24. — Circular  Revolving  Field. 

impedance  in  the  secondary  circuit,  there  is  produced  a  re- 
volving field  of  constant  intensity  when  the  rotor  revolves  at 
synchronous  speed. 

ELLIPTICAL  REVOLVING  FIELD. 

When  the  rotor  speed  is  not  truly  synchronous,  the  extremity 
of  the  vector  0  P,  which  represents  the  value  and  position  of 
the  resultant  field,  describes  an  ellipse.  For  any  given  ef- 
fective value  of  primary  field  magnetism,  the  effective  value 
of  the  induced  field  magnetism  depends  directly  upon  the  speed 
as  mentioned  above.  Let  S  represent  the  speed  with  synchronism 
as  unity,  then,  if  cos.  a  represents  the  instantaneous  value  of 
the  primary  magnetism,  SXsin.  a  equals  the  instantaneous 
value  of  the  induced  magnetism. 


THE  SINGLE-PHASE  INDUCTION  MOTOR. 


53 


Referring  now  to  Fig.  25,  after  any  time  lapse,  a, 
O  B  =  O  C  cos.   a  represents  the  instantaneous  value  of  the 

primary  magnetism. 

O  A  =  SXO  C  sin.  a  represents  the  value  of  the  secondary 
magnetism,  while 

O  P  =  \/O  B2  +  O  A2  represents  the.  value  and  position  of  the 
resultant  magnetism. 

Since,  from  the  figure,  the  distance  B  P  bears  a  constant  ratio 
of  5  to  distance  B  C,  the  locus  of  the  curve  described  by  the 
point  P  is  an  ellipse. 

The  vertical  axis  of  this  ellipse  is  determined  by  the  primary 
field,  while  the  horizontal  depends  upon  the  speed.  Above 


\ 


Fig.  25. — Elliptical  Revolving  Field. 


synchronism  the  figure  remains  an  ellipse,  having  its  major 
axis  along  the  induced  field  line.  At  synchronism  the  ellipse 
becomes  a  circle,  as  noted  above.  At  zero  speed  the  ellipse 
is  a  straight  line,  which  means  that  at  standstill  there  is  no 
quadrature  flux  and  hence  no  revolving  field.  Below  zero 
speed,  that  is,  with  reversed  rotation,  the  curve  is  yet  an  ellipse, 
the  side  which  was  previously  to  the  right  being  transferred 
to  the  left  and  vice  versa. 

STARTING  TORQUE  OF  THE  SINGLE-PHASE  MOTOR. 

For  the  above  reasons,  the  single-phase  induction  motor  has 
inherently  no  starting  torque  whatever,  but  will  accelerate 
almost  to  synchronism  if  given  an  initial  speed  in  either  direc- 


54  ALTERNATING  CURRENT  MOTORS. 

tion,  and,  when  the  torque  is  not  too  great,  will  operate  in  a 
manner  quite  similar  to  that  of  a  polyphase  induction  motor. 
Due  to  the  existence  of  the  quadrature  flux,  to  produce  which 
magnetomotive  force  must  be  supplied  by  current  in  the  pri- 
mary windings,  at  synchronism  the  magnetizing  current  for  a 
single-phase  motor  is  twice  as  great  per  phase  as  is  the  case 
when  the  same  machine  is  properly  wound  and  operated  on  a 
two-phase  circuit  of  the  same  e.m.f. ;  but  the  total  number  of 
exciting  ampere-turns  is  the  same  in  the  one  case  as  in  the 
other.  A  difference  in  the  performance  of  a  single-phase  from 
that  of  a  polyphase  motor  is  found  in  the  existence  at  no  load 
of  considerable  current  in  the  secondary  with  the  former,  while 
the  rotor  current  is  practically  negligible  with  the  latter.  These 
facts  will  be  discussed  more  fully  in  a  subsequent  chapter. 

USE  OF  "  SHADING  COILS." 

The  most  serious  defect  in  the  behavior  of  single-phase  motors 
is  in  connection  with  their  lack  of  starting  torque,  to  remedy 
which  many  ingenious  devices  have  been  developed.  A  simple 
method  of  producing  the  requisite  quadrature  magnetic  flux 
for  the  purpose  of  giving  a  single-phase  induction  motor  a 
starting  torque  is  found  in  the  use  of  "  shading  coils,"  which 
are  extensively  employed  in  alternating-current  fan  motors. 
Each  coil  consists  of  a  low  resistance  conductor  surrounding  a 
portion  of  a  field  pole.  As  ordinarily  applied,  there  is  cut  in 
each  pole  a  slot  parallel  to  the  shaft  of  the  rotor.  In  this  slot 
is  placed  the  conductor,  which  is  connected  by  a  closed  path  of 
high  conductivity  around  a  portion  of  the  pole  included  between 
the  slot  and  the  side  of  the  pole,  as  shown  in  Fig.  26.  The  coil 
of  each  pole  is  placed  similarly  to  that  of  the  other  poles,  and, 
as  explained  below,  the  secondary  revolves  in  the  direction  from 
that  portion  of  a  pole  not  surrounded  by  a  coil  towards  the 
"  shaded  "  side  of  the  pole. 

The  action  of  the  shading  coils  is  as  follows,  reference  being 
had  to  Fig.  26:  Consider  the  field  poles  to  be  energized  by 
single-phase  current,  and  assume  the  current  to  be  flowing  in 
a  direction  to  make  a  north  pole  at  the  top.  Consider  the  poles 
to  be  just  at  the  point  of  forming.  Lines  of  force  will  tend  to 
pass  downward  through  the  shading  coil  and  the  remainder 
of  the  pole.  Any  change  of  lines  within  the  shading  coil  gen- 


THE  SINGLE-PHASE  INDUCTION  MOTOR. 


55 


erates  an  e.m.f.,  which  causes  to  flow  through  the  coil  a  current 
of  a  value  depending  on  the  e.m.f.  and  always  in  a  direction 
to  oppose  the  change  of  lines.  The  field  flux  is,  therefore, 
partly  shifted  to  the  free  portion  of  the  pole,  while  the  accu- 
mulation of  lines  through  the  shading  coil  is  retarded.  How- 
ever, so  long  as  the  magnetomotive  force  of  the  field  current 
is  of  sufficient  strength  and  in  the  proper  direction,  the  lines 
through  the  shading  coil  increase  in  number,  although  the 
increase  is  retarded,  which  is  to  say,  that,  even  after  the  lines 
in  the  other  part  of  the  pole  begin  to  decrease  in  number,  the 


Primary  Magnetism 

"North"  and  Increasing 

s 

1 

i 

Primary  Magnetism 
"Zero"  and  about 

•  —  •***r 

Shading      i       || 

•WWPr 

Correot     7       II 

to  Reverse 
•  >, 

B 

li  —  ii 

Shading 
Coil 

^rt  1 

• 


FIGS.  26  and  27.— Action  of  Shading  Coils. 

flux  within  the  shading  coil  is  increasing  and  continues  to  in- 
crease till  the  exciting  current  drops  to  a  strength  just  sufficient 
to  maintain  that  density  of  flux  which  is  within  the  coil.  At 
this  instant  the  flux  in  the  shading  coil  has  its  maximum  value 
as  a  north  magnetic  pole. 

As  the  flux  on  the  other  portion  of  the  pole  continues  to  de- 
crease, the  lines  within  the  shading  coil  tend  to  decrease  also, 
but  the  e.m.f.  generated  by  their  rate  of  change  causes  to  flow 
a  current  which  tends  to  prevent  any  change  of  lines,  so  that 
when  the  other  portion  of  the  pole  contains  no  lines  whatever 
there  is  yet  within  the  shading  coil  an  appreciable  amount  of 


56  ALTERNATING  CURRENT  MOTORS. 

flux,  forming  a  north  magnetic  pole,  as  indicated  by  Fig.  27, 
the  result  being  a  shifting  of  the  field  from  the  unshaded  to  the 
shaded  side  of  each  pole.  This  is  repeated  when  the  mag- 
netism reverses.  The  lines  within  the  shading  coil  decrease, 
with  an  increasing  rate  of  change,  and  a  condition  of  zero 
lines  within  the  coil  is  soon  reached.  At  this  instant,  there  is 
a  south  magnetic  pole  at  the  "  unshaded  "  end  of  the  top 
field  pole,  and  a  north  magnetic  pole  at  the  "  unshaded  "  end 
of  each  adjacent  field  pole,  and  south  and  north  magnetic  poles 
begin  immediately  to  form  within  the  corresponding  shading  coils. 
Looking  back  over  the  process  of  formation  of  the  magnetic 
poles,  it  is  seen  that  north  poles  have  occupied  successively 
the  following  positions:  top  "  unshaded,"  top  total,  top  "shaded," 
side  "  unshaded,''  side  total,  etc.  Or,  more  simply,  the  north 
pole,  though  varying  in  strength,  has  travelled  in  a  counter 
clock-wise  direction.  This  process  is  continuous,  and  results 
in  a  truly  rotating  field.  A  rotor  placed  in  this  field  is  drawn 
into  rotation  as  though  the  primary  had  been  properly  wound 
and  connected  to  an  unsymmetrical  polyphase  circuit.  This 
method  cannot  be  satisfactorily  and  economically  applied  to 
motors  of  large  sizes. 

USE  OF  COMMUTATOR  ON  THE  ROTOR. 

A  simple  method  of  applying  a  commutator  to  induction 
motors  for  starting  purposes  is  to  utilize  current  produced  in 
the  armature  by  the  alternating  flux  from  the  field.  In  the 
application  of  this  method  the  rotor  is  provided  with  a  winding 
similar  to  that  of  a  direct-current  armature,  connected  to  a 
commutator  in  a  manner  very  similar  to  that  commonly  em- 
ployed with  direct-current  machinery.  Due  to  facts  discussed 
below,  current  which  flows  through  the  armature,  by  way  of 
suitably  connected  brushes,  gives  to  the  rotor  sufficient  torque 
to  bring  it  under  full  load  to  a  predetermined  speed  at  which  a 
mechanism  operated  by  centrifugal  force  causes  a  short-circuiting 
device  to  inter-connect  all  the  segments  of  the  commutator,  and 
thus  to  convert  the  armature  into  virtually  a  squirrel-cage  rotor. 

The  function  of  the  commutator  in  the  production  of  the  start- 
ing torque  may  be  determined  by  reference  to  Fig.  28.  Con- 
sider a  closed  coil  armature  supplied  with  a  commutator  to  be 
situated  at  rest  between  two  poles  of  a  motor,  which  poles 


THE  SINGLE-PHASE  INDUCTION  MOTOR. 


57 


are  excited  with  alternating  current,  as  indicated  in  the  drawing. 
There  will  be  generated  in  each  coil  of  the  armature  an  alter- 
nating e.m.f.  of  a  value  depending  upon  the  rate  at  which  that 
coil  is  cut  by  the  alternating  field  flux.  The  coils  which  lie  in  a 
horizontal  plane  will  be  cut  by  the  maximum  flux,  while  those 
in  the  vertical  plane  will  be  cut  by  the  minimum  flux,  which 
minimum  will  be  zero  when  the  plane  becomes  truly  vertical. 
Though  all  the  coils  are  connected  in  a  continuous  circuit,  no 
current  will  flow  in  the  armature,  since  the  e.m.f.  on  one  side 


FIG.  28. — Production  of  Starting  Torque. 

is  equal  to  that  on  the  other,  and  the  two  sides  are  in  series. 
The  maximum  e.m.f.  will  exist  between  a  top  and  a  bottom  arma- 
ture coil,  or  between  opposite  commutator  segments,  which  lie 
in  the  vertical  plane. 

Let  E  represent  the  effective  value  of  this  maximum  e.m.f., 
then  E  cos.  6  will  represent  the  value  of  the  e.m.f.  between 
opposite  commutator  segments  occupying  a  plane  forming  an 
angle  of  6  degrees  with  the  vertical. 

Consider  two  opposite  commutator  segments  to  be  connected 
together  externally  by  a  conductor  and  appropriate  brushes, 


58  ALTERNATING  CURRENT  MOTORS. 

and  represent  the  impedance  of  the  armature  and  the  external 
circuit  by  Z.  Then  a  current  will  flow  through  this  circuit, 
which  current  may  be  represented  in  value  at  any  given  position 
of  the  brushes  by  , 

_E  cos.  0 
~Z      ' 

This  current  will  be  a  maximum  when  the  brushes  occupy 
the  vertical  plane,  and  a  minimum  when  they  are  in  the  hori- 
zontal plane. 

Referring  again  to  Fig.  28,  and  remembering  that  a  conductor 
carrying  a  current  in  a  magnetic  field  experiences  a  torque 
which  is  proportional  to  the  product  of  the  current,  the  field 
and  the  cosine  of  the  angle  between  them,  it  will  be  seen  that, 
for  a  given  current  in  the  armature,  the  maximum  torque  would 
be  exerted  when  the  brushes  are  in  the  horizontal  plane,  and 
that  the  armature  will  experience  no  torque  whatever  when  the 
brushes  are  in  the  vertical  plane.  It  is  to  be  noted,  however, 
that  when  the  brushes  are  in  the  horizontal  plane  no  current 
will  be  produced  in  the  brush  circuit,  and,  therefore,  the  torque 
is  zero.  In  any  intermediate  position  the  current  in  the  brush 
circuit  gives  a  certain  torque. 

Obviously,  when  the  current  and  flux  reverse  together  the 
torque  continues  to  be  exerted  in  one  direction  and  the  rotor 
is  given  the  desired  initial  speed  previous  to  being  converted 
to  a  squirrel  cage  rotor,  as  stated  above. 

Single-phase  motors  equipped  with  starting  devices  of  this 
nature  give  satisfactory  results  as  to  simplicity  of  operating 
circuits,  efficiency  of  performance  and  reliability  of  service 
quite  comparable  to  those  obtained  with  polyphase  motors 
This  type  of  machine  in  its  starting  condition  is  frequently 
referred  to  as  a  "  repulsion  "  motor.  The  repulsion  motor  is 
treated  at  great  length  both  graphically  and  algebraically  in 
subsequent  chapters. 

POLYPHASE   INDUCTION   MOTORS   USED   AS   SINGLE-PHASE   MA- 
CHINES. 

Induction  motors  are  frequently  started  up  from  rest  on 
single-phase  circuits  by  operating  them  as  so-called  "  split-phase" 
machines.  Commercially  considered,  a  split-phase  motor  is  a 
polyphase  machine,  the  current  in  the  separate  phases  being 


THE  SINGLE-PHASE  INDUCTION  MOTOR. 


59 


obtained  at  different  lag  angles  from  a  single-phase  circuit,  so 
that  a  starting  torque  is  produced  at  the  rotor. 

A  two-phase  induction  motor  may  be  brought  up  to  speed 
on  a  single-phase  circuit  by  connecting  the  windings  of  both 
phases  to  the  circuit,  one  directly  and  the  other  through  a  suit- 
ably chosen  resistance  or  condensance,  as  shown  in  Fig.  29. 
The  current  through  the  circuit  containing  the  resistance  will 


FIG.  29. — Circuits  of  (Two-phase) 
Single-phase  Motor. 


Phase  A 

FIG.  30. — Circuits  of  (Three-phase) 
Single-phase  Motor. 


alternate  more  nearly  in  unison  with  the  impressed  e.m.f.  than 
that  in  the  other,  and  it  will  possess  a  component  in  quadrature 
to  the  current  in  the  other  circuit,  which  component  will  pro- 
duce the  desired  quadrature  flux.  The  elliptical  revolving  field 
thus  produced  will  give  a  torque  which  will  start  the  motor  up 
from  rest,  if  the  load  be  not  too  great.  When  about  half  speed 
has  been  attained  the  circuit  through  the  resistance  is  cut  out 
and  the  motor  operates  as  a  single-phase  machine. 

Under  the  conditions  of  operations,  it  will  be  found  that  there 


60 


ALTERNATING  CURRENT  MOTORS. 


is  generated  in  the  inactive  phase  winding  an  e.m.f.  equal  (for 
negligible  secondary  impedance)  to  the  counter  e.m.f.  of  mechan- 
ical motion  in  the  active  phase  winding,  and  that  this  e.m.f. 
is  almost  in  quadrature  in  time-position  with  the  supply  e.m  f. 
The  lack  of  exact  quadrature  is  due  to  the  lag  of  the  counter 
e.m.f.  of  rotation  in  the  active  coil  behind  the  impressed  e.m.f. 
and  to  some  extent  to  the  further  lag  of  the  secondary  exciting 
current  behind  this  e.m.f.,  and  the  sign  of  the  angle  of  dis- 

B, 


FIG.  31. — Diagram  of  Two-phase  FIG.  32. — Diagram  of  Three-phase 


e.m.f. 's. 


e.m.f.'s. 


placement  from  90°  depends  upon  the  direction  of  rotation  of 
the  motor  secondary.  Fig.  31  shows  the  relative  value  and 
position  of  this  tertiary  e.m.f.  for  a  two-phase  motor  running 
single-phase,  while  Fig.  32  indicates  equivalent  results  for  a 
three-phase  motor  on  a  single-phase  circuit. 

By  combining  the  e.m.f.  of  the  inactive  winding  of  a  two- 
phase  motor  with  that  of  the  supply  circuit,  there  is  available 
an  almost  symmetrical  two-phase  circuit,  from  which  may  be 
started  at  once,  without  auxiliary  apparatus,  any  similar  two- 
phase,  or,  by  a  few  slight  changes,  any  three-phase  induction 


THE  SINGLE-PHASE  INDUCTION  MOTOR. 


61 


motor.  The  draught  of  current  from  the  inactive  phase  winding 
will  have  very  little  effect  upon  the  operation  of  the  first  motor. 
By  this  method  it  is  possible  to  dispense  with  auxiliary  starting 
apparatus  for  all  motors  of  any  given  installation,  with  the  ex- 
ception of  one,  and,  with  properly  arranged  circuits,  two-phase 
motors  may  in  this  manner  be  started  up  with  a  fair  operating 
torque. 

A  three-phase  induction  motor  may  be  operated  from  a  single- 
phase  circuit  by  connecting  two  leads  from  the  motor  directly 
to  the  supply  circuit  and  joining  the  third  to  an  auxiliary 
starting  circuit,  formed  by  placing  a  resistance  and  a  reactance 
in  series  across  the  supply  circuit,  as  indicated  by  Fig.  30. 

The  effect  of  placing  the  resistance  and  reactance  in  series 


•~  B  A  B 

FIGS.  33  and  34. — Vector  Diagram  of  Electromotive  Forces. 

is  to  displace  the  relative  potential  of  the  point  where  the  two 
join,  from  a  line  connecting  the  extremities  of  the  two,  as  shown 
in  Fig.  33.  For  a  true  reactance  the  locus  of  this  point,  with 
varying  resistance,  is  the  arc  of  a  circle,  as  indicated  by  Fig.  34, 
and  the  maximum  displacement  occurs  when  the  resistance 
and  reactance  are  equal.  Under  this  condition  the  displacement 
(when  no  current  is  being  taken  off  at  P)  will  be  .5  £T,  or  one- 
half  the  line  voltage.  For  a  true  three-phase  circuit,  the  dis- 
placement should  be 

\/3 

~2~  ET  =  .866  ET. 

It  is  clear,  therefore,  that  this  method  cannot  possibly  give  e.m.fs. 
in  true  three-phase  relation.  It  is  found,  however,  that  the 


62  ALTERNATING  CURRENT  MOTORS. 

displacement  obtained  is  adequate  for  starting  motors  of  mod- 
erate sizes. 

The  use  of  condensance,  instead  of  inductance,  offers  some 
advantages,  since  by  properly  proportioning  the  condensance, 
the  leading  current  demanded  may  be  adjusted  to  equality 
with  the  lagging  exciting  current  during  operation  and,  theoret- 
ically, a  power  factor  of  unity  may  be  obtained.  The  disturb- 
ing influence  of  change  of  frequency,  the  compensating  dis- 
advantages due  to  presence  of  higher  harmonics  from  the 
distortion  of  the  e.m.f.  wave  from  a  true  sine  curve  of  time- 
value,  and  the  practical  necessity  of  operating  condensers  at 
high  voltage,  coupled  with  the  lack  of  satisfactory  commercial 
condensers  in  convenient  form,  have  limited  the  application  of 
this  method. 


CHAPTER  VII. 
GRAPHICAL  TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA. 

ADVANTAGE  OF  GRAPHICAL  METHODS. 

With  almost  no  exception,  the  graphical  method  of  treatment 
of  electrical  phenomena  does  not  produce  results  as  accurate 
as  the  analytical;  yet,  for  many  purposes  where  a  fine  degree 
of  accuracy  is  not  required,  the  ease  of  manipulating  has  led  to 
the  extensive  use  of  the  graphical  method,  approximate  results 
being  first  rapidly  determined,  after  which  if  greater  accuracy 
is  desired,  the  analytical  method  may  be  used.  In  cases  where 
only  qualitative  results  are  desired,  but  where  some  knowledge 
of  the  effect  of  change  in  the  different  variables  connected  with 
the  phenomena  is  important,  the  graphical  method,  because  of 
its  simplicity,  readily  lends  itself  to  the  quick  determination 
of  results  sufficiently  accurate  for  the  needs  of  the  case. 

Before  discussing  the  graphical  diagram  for  the  represent"4"  ,n 
of  the  value  and  phase  of  primary  and  secondary  current  ,  of 
an  induction  motor,  it  is  well  to  establish  the  similarity  between 
an  induction  motor  and  a  static  transformer. 

EFFECT  OF  INSERTING  RESISTANCE  IN  THE  SECONDARY. 

At  any  value  of  field  magnetism,  the  effect  of  inserting  re- 
sistance in  the  secondary  of  an  induction  motor,  for  a  given 
value  of  secondary  current,  is  to  vary  the  slip  directly  with  the 
total  secondary  resistance  without  affecting  either  the  torque 
or  the  power-factor. 

As  proof  of  this  fact,  let  E2  =  the  e.m.f.  which  would  be 
generated  in  the  secondary  at  100  per  cent,  slip,  at  the  given 
field  magnetism. 

5  =  slip,  with  synchronism  as  unity, 
7? 2  =  secondary  resistance, 

X2  =  secondary  reactance  at  100  per  cent,  slip  (standstill ;  5  =  1) 
7  2  =  secondary  current. 

63 


64  ALTERNATING  CURRENT  MOTORS, 

Then 

_          s£2  2  ^         s2E22 

"  °       = 


or,  since  72  and  E2  are  constant  for  the  chosen  condition  of  service, 
K*  -          S* 


/ K2 

\I  -  K2  X* 


^  IT  T^O     ~\.7     o 


f^2    ...    ^2   7?  2 
2 


or  the  slip  is  directly  proportional  to  the  secondary  resistance. 
The  secondary  power-factor,  cos.  6  = 

R2 


R 


Vl  +a2  X22 

or  the  secondary  power-factor  is  independent  of  the  secondary 
resistance. 

The  total  secondary  power  is  72  E2  cos.  0,  and  is  independent 
of  the  secondary  resistance,  while  the  torque,  which  is 

7.04  —  -  —  2      ,  ,  is  also  independent  of  the  secondary  resistance. 
syn.  speed 

Since  the  effect  of  inserting  resistance  in  the  secondary  cir- 
cuit is  merely  to  increase  the  slip  without  altering  the  other 
qualities,  it  follows  that  by  using  suitable  selected  resistances, 
the  slip  can  be  made  unity  for  any  value  of  secondary  current 
at  the  corresponding  power-factor.  In  consequence  of  this 
fact,  the  performance  of  the  secondary  will  be  faithfully  repre- 
sented if  all  the  power  received  from  the  primary  be  considered 
as  dissipated  in  resistance  in  the  secondary  circuit,  the  slip  at 
all  times  being  taken  as  unity  and  the  total  secondary  resist- 
ance (conductance)  being  assumed  to  be  varied  according  to 
the  secondary  load,  or  briefly,  the  induction  motor  may  be 
treated  in  all  respects  like  a  stationary  transformer.  The 
determination  of  the  slip,  torque,  etc.,  of  the  motor  under  oper- 
ating conditions  will  be  discussed  later. 


TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA.     65 

PRIMARY  AND  SECONDARY  CURRENT  Locus. 

Perhaps,  of  all  the  graphical  diagrams  which  have  been  sug- 
gested at  various  times  to  represent  the  performance  of  an 
induction  motor,  the  simplest,  and  at  the  same  time  the  most 
complete,  is  that  showing  the  value  and  phase  of  the  primary 
and  secondary  currents.  The  quantities  intended  to  be  repre- 
sented by  this  diagram  can  best  be  ascertained  by  investigating 
the  method  of  determining  the  points  on  the  current  locus, 
such  as  is  shown  in  Fig.  35,  which  is  plotted  according  to  the 
following  instructions: 

Use  the  vertical  scale  (at  a  certain  number  of  amperes  per 


150 

1 

g     ion 

^< 

^ 

^-^" 

•^> 

F 

3      19fl 

x^ 

O     Jl^U 

n_i      -|-«n 

/ 

o     H° 
a    100 
I      9° 

/ 

/ 

/ 

a    8° 

o      ,-»/\ 

-H— 

Q 

^ 

0      70 

JH           ftfl 

/// 

o      *»n 

/ 

77 

P     50 

V 

j 

// 

oT     w 

// 

7 

/ 

s   ** 

^ 

/ 

// 

a    ^ 

^      10 

0 

/ 

/ 

c 

.  — 

a 

==" 

) 

JA 

4 

) 

6 

0 

~8 

3 

~To 

0 

12 

0~" 

^4 

0 

16 

0 

It 

>0 

2C 

0 

~£i 

0~ 

240 

Amperes,  Wattless  Component  of  Current 
FIG.  35. — Primary  and  Secondary  Current  Locus. 

inch)  to  plot  values  of  the  power  component  of  the  currents 
(7  cos.  6),  and  the  horizontal  scale  (at  the  same  number  of  am- 
peres per  inch)  to  plot  the  wattless  component  of  the  currents 
(7sin#). 

From  the  origin  0,  lay  off  a  distance  O  B  equal  to  the  power 
component  of  the  primary  current  at  no  load,  and  from  the 
point  B  draw  the  horizontal  line  B  A  with  a  value  equal  to 
that  of  the  wattless  component  of  the  primary  current  at  no 
load.  The  line  O  A  represents  the  no  load  primary  current, 
while  the  angle  A  O  B  is  the  primary  angle  of  lag  at  no  load. 
In  a  similar  manner,  selecting  any  load  current,  as  0  C,  lay  off 


66  ALTERNATING  CURRENT  MOTORS. 

the  power  and  wattless  components  O  D  and  D  C  The  angle 
of  lag  at  this  load  is  represented  by  the  angle  COP.  Follow- 
ing out  the  same  method,  locate  a  number  of  points  corre- 
sponding to  the  points  A  and  C,  and  draw  a  smooth  curve 
through  them.  At  any  point  on  this  curve,  as  P, 

O  P  represents  the  primary  current, 

PR  "     wattless  component  of  the  primary  current, 

OR  ''     power  component  of  the  primary  current, 

P  O  R  "     primary  angle  of  lag. 

P  Q  wattless  component  of  the  secondary  cur- 

rent. 

A  Q  "     power  component  of  the  secondary  current 

A  P  "     secondary  current, 

PAG  "     secondary  angle  of  lag. 

The  proof  of  the  representation  of  the  secondary  quantities 
is  as  follows:  When  running  under  load,  B  R  equals  the  increase 
in  the  power  component  of  the  primary  current  over  its  no  load 
value.  As  in  a  transformer,  this  increase  is  due  to  the  flow 
of  current  in  the  secondary,  and  being  in  phase  (opposition) 
to  the  power  component  of  the  secondary  current,  is  a  direct 
measure  of  its  value.  A  similar  course  of  reasoning  holds  for 
the  wattless  component  P  Q.  Hence  A  P  represents  the  sec- 
ondary current  both  in  value  and  phase  position. 

An  inspection  of  the  diagram  will  show  that  the  maximum 
secondary  power  factor  occurs  at  no  load,  while  the  maximum 
primary  power  factor  occurs  at  that  load  which  causes  the 
line  O  P  to  become  tangent  to  the  curve  A  C  P  F.  Since  the 
no  load  losses  are  equal  to  the  product  of  O  B  by  the  primary 
e.m.f.,  and  the  secondary  current,  primary  current  and  power 
factor  can  be  obtained  directly  from  the  curve  for  any  value  of 
primary  load,  it  follows  that  if  the  primary  and  secondary  re- 
sistances be  known,  the  input  losses,  output,  efficiency,  slip, 
speed,  power-factor,  apparent  efficiency,  and  torque  can  easily 
be  determined  when  the  curve  A  C  P  F  is  located. 

TEST  RESULTS. 

The  accompanying  table  records  results  of  calculations  made  for 
a  10-h.p.,  two-phase,  220- volt  induction  motor,  use  being  made  of 
Fig.  35.  which  has  been  constructed  in  part  from  data  obtained 
with  this  machine.  Results  found  in  the  table  have  been  plotted 


TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA 

O  O  35  CO  Cl       CO  O  3i  ">O  CN 
CO 

a 
8 
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•IUSIUO.II{3 

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"(ill's 


•sso[  aaddoo 


«fl  CO  CO  00  N-       00  <N  >O  '-O  M  tt  <N  O  O 


§O5X>r>-iOTt<'M  —  O5t>->OMXCOO'*C7SCOO 
053535050iC3>a>XC«»Xt!--^t--O^^HO 


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«O  CO  CC  --O  «O  O5  00  N  00  ^        •*  i«  -<  CO 

XOiMCC'f^M<rl1<WOt^-^C<>-HO 

—  n  t  i.o  'O  i--  x  o  o  "H  —i  IN  s^i  c^  FH 


uo;oui  o;  induj 


o»ooco-<cooocococo*-i-ico;o«ococococo 

<M>O-^*XOO»OX)  —  -^COO5t>-r>-t>.^Ht>-t>.COO5 


>  O  O  O  O  O  O 


§r}<  -t  iO  O  O  --0  t^  X  O  C<l  »O  O  t^  00  -i  CO  00  c 
i-iclo5^i35«Jt»35ot-«o5i*i55t**i-<< 


jo 


68 


ALTERNATING  CURRENT  MOTORS. 


in  the  form  of  the  curves  shown  in  Figs.  36  and  37,  from  which 
can  be  ascertained  the  complete  performance  of  the  motor 
from  no  load  to  full  load  and  beyond.  This  motor  is  supplied 
with  a  variable  resistance  external  to  the  secondary  windings 


0123456789 
Horse  Power  Output 


10 


4     5     6     7     8     9    10    11   12  13  14    15   16  17   18  19  20   &., 
Horse  Power  Output 

FIGS.  36  and  37. — Performance  of  Two-phase  Induction  Motor. 


for  the  purpose  of  decreasing  the  primary  current  and  increasing 
the  rotor  torque  during  the  starting  period.  The  effect  of 
using  this  resistance  is  seen  at  a  glance  when  Fig.  36  is  com- 
pared with  Fig.  37. 


TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA.     69 

From  column  9  of  the  table  it  is  seen  that  the  maximum  torque 
of  the  rotor  occurs  at  that  value  of  the  primary  current  which 
allows  the  greatest  amount  of  power  to  be  delivered  to  the 
secondary  and  that  the  value  of  this  torque  and  the  primary 
current  and  power-factor  at  which  it  occurs  are  entirely  indepen- 
dent of  the  secondary  resistance.  Columns  10  and  11  of  the 
same  table,  however,  show  that  the  slip  at  which  occurs  the 
maximum  torque  is  directly  proportional  to  the  secondary  re- 
sistance, though  the  secondary  current  which  gives  this  torque 
has  a  definite  fixed  value  independent  of  both  the  slip  and  the 
secondary  resistance.  It  is  evident,  therefore,  that  by  the  use 
of  variable  resistance  in  the  secondary  circuit,  the  maximum 
torque  can  be  made  to  occur  at  any  speed  from  standstill  to  a 
few  per  cent,  below  synchronism,  but  that  the  primary  current 
will  depend  upon  the  torque  and  not  upon  the  speed. 

While  the  method  of  calculation  used  in  arriving  at  the  re- 
sults given  in  each  column  of  the  table  is  explained  at  the  heads 
of  the  separate  columns,  perhaps  a  few  words  should  be  added 
concerning  the  formulas  for  determining  the  slip  and  the  torque. 
The  rotor  slip  may  be  expressed  as  the  ratio  of  the  copper  loss 
of  the  secondary  to  the  total  power  received  from  the  primary. 
That  is, 

S==H_RI 

where  W 's  is  the  total  secondary  power.  This  relation  was  dis- 
cussed at  length  in  a  previous  chapter,  and  it  need  not  be 
dwelt  upon  at  this  place.  Likewise  it  has  been  shown  that 
the  rotor  torque  may  be  expressed  in  pounds  at  one  foot  radius 
by  the  formula, 

D  ^    7.04  W, 

syn.  speed' 

It  will  be  observed  from  column  7,  that  there  has  been 
added  to  the  primary  copper  loss  a  quantity,  856  watts,  to  obtain, 
the  total  primary  loss.  The  856  watts  is  the  so-called  "  con- 
stant loss,"  and  includes  all  iron  and  friction  losses  of  the 
motor.  It  should  be  carefully  noted  in  this  connection  that 
this  loss  is  not  constant,  but  it  varies  with  increase  of  load. 
There  are  three  causes  which  affect  the  constancy  of  the  value  of 
the  iron  loss.  The  drop  in  e.m.f.  due  to  the  current  through  the 


70  ALTERNATING  CURRENT  MOTORS. 

primary  resistance  lessens  the  e.m.f.  to  be  balanced  by  the  rate 
of  change  of  the  core  flux  and  thus  decreases  the  density  of  mag- 
netism and  tends,  thereby,  to  reduce  the  iron  loss.  When  the  rotor 
travels  at  synchronous  speed  the  secondary  core  experiences  no 
reversal  in  magnetism  and  there  is,  therefore,  no  secondary  iron 
loss.  As  load  is  placed  upon  the  motor,  the  rotor  speed  de- 
creases and  the  magnetism  of  the  secondary  core  reverses  at  a 
rate  proportional  to  the  slip  and  there  is  produced  a  correspond- 
ing core  loss,  tending  to  increase  the  total  iron  loss  of  the  motor. 
Another  cause  tending  to  increase  the  iron  loss  of  both  the 
primary  and  secondary  cores  is  the  loss  in  the  iron  immediately 
around  each  conductor  due  to  the  superposed  local  flux  when 
current  traverses  the  conductor.  The  flux  causing  this  local 
loss  is  that  to  which  is  due  the  reactance  of  the  primary  and 
secondary  coils.  The  value  of  this  flux  depends  directly  upon 
the  current  in  the  conductors  and  inversely  upon  the  total 
reluctance  of  the  magnetic  path  surrounding  them.  The  loss 
due  to  a  given  flux  in  a  certain  mass  of  iron  depends  not  only 
upon  the  value  of  the  flux  and  the  frequency  of  its  reversal 
but  to  a  large  extent  also  upon  the  density  of  magnetism  upon 
which  this  flux  is  superposed.  In  any  commercial  motor  the 
density  of  magnetism  in  the  core  teeth  is  normally  quite  high 
on  account  of  the  field  magnetism  proper  alone,  so  that  the  loss 
due  to  the  local  flux  surrounding  each  slot  in  both  the  primary 
and  secondary  cores  under  load  currents  depends  greatly  upon 
other  factors  than  its  own  value  and  periodicity  of  reversal. 
The  increase  of  iron  loss  due  to  the  second  and  third  causes, 
as  enumerated  above,  is  ordinarily  somewhat  greater  than  the 
decrease  due  to  the  first  cause,  resulting  in  the  so-called  load 
losses.  In  comparison  with  the  total  losses  the  increase  in  iron 
loss  is  usually  quite  small  throughout  the  operating  range  of 
the  motor.  When  the  character  of  the  work  necessitates  such 
procedure,  this  increase  can  be  approximately  determined  and 
corresponding  corrections  made,  though  other  errors  in  com- 
putations or  assumptions  will  frequently  more  than  equal  that 
due  to  the  neglect  of  this  increase. 

EQUATION  OF  THE  CURRENT  Locus. 

Referring  now  to  Fig.  35,  it  will  be  seen  that  the  curve  A  C  P  F, 
the  current  locus,  has  been  drawn  as  the  arc  of  a  circle.     This 


TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA,     71 

is  the  construction  usually  adopted  for  the  purposes  of  pre- 
liminary design,  in  which  case  the  point  F  is  located  as  the 
extremity  of  the  vector  representing  the  value  and  phase  of  the 
primary  current,  which  would  be  obtained  with  full  e.m.f. 
impressed  upon  the  motor  with  the  rotor  stationary.  This 
construction  is  partly  justified  by  the  following  fact: 

If  the  primary  resistance  and  the  exciting  current  be  neglected, 
or  the  secondary  current  be  considered  equal  to  the  primary, 
the  current  locus  is  a  true  circle. 
Let  E  =  impressed  e.m.f., 

7  =  current  to  the  motor, 
Xi  =  primary  reactance, 
X2  =  secondary  reactance, 
R2  =  secondary  resistance. 

Treating  the  induction  motor  as  a  transformer,  the  value  of 
the  current  as  R^  is  varied  is 

"  /-  ^==^= 

but  R2  =  (X1+X2)  cotand;  hence  7  = 

E 


sin  6 

which,  when  E  is  constant,  is  the  polar  equation  of  a  circle 
having  a  diameter  of  E-r-(Xl-\-X2). 

ERRORS  IN  ASSUMING  A  CIRCULAR  ARC. 

If  the  primary  coils  carried  at  all  times  an  exciting  current  of 
constant  value  in  addition  to  a  current  equal  and  opposite  to 
the  secondary  current,  the  curve  would  yet  be  a  true  circle. 
In  this  case,  however,  the  primary  current  would  need  to  be 
measured  from  a  point  external  to  the  circle  at  a  distance  from 
the  circumference  equal  to  the  exciting  current,  that  is,  the 
primary  current  would  be  measured  from  B  while  the  secondary 
current  would,  as  formerly,  be  measured  from  A. 

If  in  addition  to  the  exciting  current  the  primary  coils  carried 
a  power  component  of  current  for  the  no-load  iron  losses,  etc., 
over  and  above  the  counter  current  to  oppose  the  current  in 
the  secondary,  the  locus  would  remain  a  circle,  as  before,  but 
the  primary  current  would  be  measured  from  a  point  O  so 
located  that  O  B  equals  the  power  component  of  the  no-load 


72  ALTERNATING  CURRENT  MOTORS. 

current,  Fig.  35.  The  last  assumption  is  true  for  any  induction 
motor  in  which  the  drop  in  e.m.f.  due  to  the  passage  of  the 
current  through  the  local  impedance  of  the  primary  coils  is 
negligible.  The  local  primary  impedance  is,  however,  not 
negligible  in  a  commercial  motor.  It  is  found,  however,  that  the 
error  in  determining  the  primary  and  secondary  currents  due  to 
the  slight  deviation  of  the  curve  from  a  true  circle  between  the 
points  A  and  5  produces  an  almost  inappreciable  effect  upon  the 
results. 

The  effect  of  the  primary  impedance  is  the  same  in  all  re- 
spects as  though  the  primary  coils  were  without  impedance  and 
the  power  supplied  to  the  motor  were  transmitted  over  a  circuit 
having  an  impedance  equal  to  that  of  the  primary,  which  means 
that  in  the  equation,  E  decreases  with  an  increase  of  /,  and 
the  equation  is,  therefore,  not  that  of  a  true  circle.  In  a  sub- 
sequent chapter  these  facts  will  be  discussed  more  fully. 

Having  drawn  attention  to  these  errors  in  the  assumptions 
concerning  the  current  locus,  it  is  sufficient  here  to  state  that 
while  the  method  used  does  not  give  the  absolutely  true  loca- 
tion of  the  primary  current  curve  throughout  its  whole  length, 
and  the  absolutely  true  secondary  current  curve  does  not 
coincide  with  that  of  the  primary,  the  errors  introduced  into 
the  calculations  are  relatively  small  and  for  most  practical 
purposes  may  well  be  neglected. 

The  product  of  the  power  component  of  the  primary  current 
by  the  circuit  e.m.f.  gives  the  input  to  the  motor.  It  will  be 
seen  from  Fig.  35  that  the  maximum  power  which  it  is  possible 
for  the  motor  to  receive  is  determined  by  the  radius  of  the 

E2 
circle  APF,  that  is,  it  is  represented  by  the  quantity 


*  ( 

(neglecting  the  no-load  losses),  or  the  maximum  power  which 
the  motor  can  receive  is  determined  wholly  by  the  reactance 
of  the  primary  and  secondary  coils. 

A  further  inspection  of  Fig.  35  will  show  that  the  maximum 
power  factor  occurs  at  that  value  of  primary  current  which 
causes  the  line  0  P  to  become  tangent  to  the  curve  A  P  F. 
The  value  of  this  power  factor  depends  upon  the  radius, 


TREATMENT  OF  INDUCTION  MOTOR  PHENOMENA      73 

of   the    arc   A  P  F   and  upon  the  exciting  current  A  B,  being 
increased  as  the  former  is  increased  or  as  the  latter  is  decreased. 

EFFECT  OF  DESIGN  ON  LEAKAGE  REACTANCE. 

It  is  evident  from  the  foregoing  that  in  the  construction  of 
induction  motors  every  effort  should  be  made  to  render  the 
reactance  of  the  coils  as  small  as  possible.  The  reactance  flux 
can  be  decreased  by  increasing  the  reluctance  of  the  path 
which  it  must  travel,  that  is  by  using  open  slots  and  operating 
the  core  teeth  at  high  magnetic  density.  A  method  of  de- 
creasing the  effectiveness  of  the  local  flux  without,  however, 
lessening  appreciably  the  number  of  lines  is  found  in  the  use  of 
many  rather  than  few  slots.  Thus  when  the  conductors  are 
bunched  in  a  single  slot  each  conductor  is  cut  by  the  lines 
due  both  to  its  own  current  and  to  that  of  its  neighbors  in 
the  same  slot,  so  that  the  reactive  e.m.f.  per  conductor  depends 
directly  upon  the  number  of  conductors  in  each  slot,  or  the 
reactive  e.m.f.  per  slot  varies  with  the  square  of  the  conductors 
therein,  and  the  total  reactance  of  a  certain  winding  is  in- 
versely proportional  to  the  number  of  slots  in  which  the  con- 
ductors are  placed. 

The  value  of  the  exciting  watts  depends  almost  entirely 
upon  the  radial  depth  of  the  air-gap  (more  properly,  upon  the 
volume  of  the  air-gap)  and  the  employment  of  a  small  air-gap 
is  desirable  for  large  operating  power  factor.  An  inspection 
of  Fig  35  will,  however,  reveal  the  interesting  fact  that  since 
the  radius  of  the  arc  A  P  F  is  independent  of  the  distance 
A  B,  the  maximum  power  of  the  motor  is  unaffected  by  the 
value  of  the  air-gap  except  as  a  change  in  the  latter  may  affect 
the  reactance  of  the  coils.  Since  a  reduction  in  the  air-gap 
is  accompanied  with  an  increase  in  the  local  reactance  flux 
around  the  coils,  one  is  led  to  the  highly  interesting  conclusion 
that  reducing  the  air-gap  of  a  given  motor  actually  decreases 
the  maximum  power  of  the  machine. 


CHAPTER  VIII. 
INDUCTION  MOTORS  AS  ASYNCHRONOUS  GENERATORS. 

OPERATION  BELOW  SYNCHRONISM. 

The  current  demanded  from  the  supply  system  by  an  in- 
duction motor  when  operating  without  load  near  synchronism 
is  found  to  consist  of  two  components,  one  in  phase,  and  the 
other  in  quadrature  with  the  impressed  electromotive  force. 
The  inphase,  power  component  is  found  as  that  value  of  amperes 
which  multiplied  by  the  impressed  volts  will  give  the  watts 
necessary  to  supply  the  internal  losses  of  the  machine,  while 
the  quadrature  component  is  found  as  that  value  of  amperes 
which  multiplied  by  the  number  of  turns  of  the  primary  wind- 
ing will  give  the  magnetomotive  force  necessary  to  cause  to 
flow  through  the  reluctance  of  their  paths  the  lines  required 
to  produce  by  their  rate  of  change  an  internal  counter 
e.m.f.  less  than  the  impressed  by  an  amount  such  as  to 
allow  the. primary  current  to  flow  through  the  impedance  of 
the  coil. 

When  the  speed  is  less  than  synchronism,  the  secondary 
windings  cut  the  revolving  field  at  a  rate  proportioned  to  the 
slip  and  the  e.m.f.  thus  generated  causes  to  flow  through  the 
secondary  conductors  a  current,  which  being  practically  in 
time-phase  with  the  magnetism  and  in  mechanically  the  same 
position  in  space,  will  give  to  the  rotor  a  torque  in  a  direction 
to  lessen  the  secondary  current,  that  is,  a  torque  tending  to 
accelerate  the  rotor.  This  current  in  its  effect  upon  the  pri- 
mary acts  as  though  it  flowed  in  the  secondary  circuit  of  a 
stationary  transformer  and  thus  requires  in  the  primary  coil  a 
current  equal  and  opposite  to  it  in  magnetomotive  force,  so 
that  the  core  magnetism  is  but  slightly  altered  by  its  presence. 
The  counter  current  appears  as  an  addition  to  the  power  com- 
ponent of  the  primary  current  and  lepresents  the  increase  in 
electrical  power  supplied  to  the  motor. 

74 


INDUCTION  MOTORS  AS  GENERATORS.  75 

OPERATION  ABOVE  SYNCHRONISM. 

When  the  rotor  is  driven  above  synchronism,  the  windings 
of  the  secondary  cut  the  synchronously  moving  magnetism  in 
a  direction  to  generate  an  e.m.f.  causing  a  current  to  flow  in  the 
secondary  conductors  in  a  direction  opposite  to  that  described 
above,  so  that  the  additional  component  of  current  required 
in  the  primary  coil  is  reversed  from  its  former  time-phase 
position  with  respect  to  the  field  magnetism  and  to  the  im- 
pressed primary  e.m.f.,  and  thus  tends  to  represent  power 
flowing  from  the  motor  and,  as  will  be  discussed  in  detail  later, 
at  a  speed  sufficiently  far  above  synchronism,  an  induction  motor 
acts  as  a  generator,  the  power  delivered  by  it  depending  upon 
the  relative  motion  of  the  secondary  windings  and  the  field 
magnetism,  that  is  to  say,  upon  the  slip  above  synchronism. 
This  characteristic  of  the  machine  has  been  frequently  availed 
of  upon  mountain  roads,  where,  in  descending,  the  primary 
circuit  of  the  motor  is  connected  directly  to  the  trolley  line 
and  the  speed  of  the  rotor  held  practically  constant  at  a  few 
per  cent,  above  synchronism,  or  increased  to  any  desired  value 
by  the  insertion  of  resistance  in  the  secondary  circuit.  Where 
the  system  of  tandem  speed  control  of  two  similar  motors  is 
used,  a  breaking  and  energy  -restoring  effect  may  be  produced, 
even  upon  level  roads,  by  placing  the  motors  in  tandem  con- 
nection while  they  are  yet  traveling  at  a  speed  between  one- 
half  and  full  synchronism,  and  the  rate  of  retardation  may  be 
rendered  quite  uniform  by  judicious  use  of  the  starting  resist- 
ance. The  same  remarks  obviously  apply  when  the  speed  con- 
trol is  obtained  by  change  in  the  number  of  magnetic  poles  of 
a  single  motor. 

VECTOR  DIAGRAM  OF  CURRENTS. 

The  relative  values  and  phase  positions  of  the  primary  and 
secondary  currents  of  an  induction  motor  at  all  speeds  both 
below  and  above  synchronism  may  be  graphically  represented 
by  a  simple  diagram  which  lends  itself  to  a  ready  interpretation 
of  the  interdepend  an  ce  of  the  various  characteristics  of  such  a 
machine.  Fig.  38  gives  a  vector  diagram  of  the  primary  cur- 
rent of  a  certain  three-phase,  asynchronous  machine  operating 
under  a  constant  impressed  e.m.f.  of  220  volts,  and  from  this 
diagram  may  be  determined  the  secondary  current  and  all  of 


76 


ALTERNATING  CURRENT  MOTORS. 


the  performance  characteristics  of  the  machine  and  thus  may 
its  behavior  as  an  asynchronous  generator  conveniently  be  in- 
vestigated. In  Fig.  38  the  distance  0  B  represents  the  power 
component  of  the  primary  current  at  no-load,  the  distance  ^A  B 
is  the  wattless  component  of  the  no-load  current,  while  O  A 
represents  both  in  value  and  phase  position  the  total  current 

Wattless  Component  of  Curretft 
10    30  30   40    50   60    70  80   90  100 


>    1XU 

X 

^ 

er  Component  of  Current-Pos 

38SSS£g88S 

4 

/ 

I 

/ 

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/ 

i 
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/ 

1 

/ 

7 

RM 

1 

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Component  of  Current-Nega'1 
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FIG.  38.—  Cu 

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10   20   30  40   50  60   70  80   90  100110 

Wattless  Component  of  Current 

rrent  Locus  of  Asynchronous  Machine. 

taken  by  the  motor  at  no-load.  The  product  of  O  B  with  the 
impressed  e.m.f.  gives  the  no-load  losses  of  the  motor,  while 
the  ratio  of  O  B  to  0  A,  the  cosine  of  the  angle  A  O  B,  is  the 
primary  power-factor  under  no-load  conditions.  At  a  certain 
slip,  the  primary  current  will  increase  to  some  value  and  phase 
such  as  is  represented  by  O  P,  of  which  O  R  is  the  power,  and 
JTP",  the  reactive  component,  respectively. 


INDUCTION  MOTORS  AS  GENERATORS.  11, 

Now,  the  increase  in  the  two  components  of  the  primary  cur- 
rent under  load  is  due  to  the  corresponding  components  of  the 
secondary  current,  so  that  such  increase  serves  as  a  measure  of 
the  secondary  current.  An  inspection  of  Fig.  38  will  show  that 
the  line  A  P  is  the  vector  sum  of  the  changes  in  the  two  com- 
ponents of  the  primary  current  over  the  no-load  values  and 
hence  represents  both  the  value  and  phase  (opposition)  position 
of  the  secondary  current  when  reduced  to  primary  terms  by 
the  inverse  ratio  of  turns  of  the  respective  windings.  A  further 
inspection  and  study  of  Fig.  38  will  show  that  a  series  of  points, 
such  as  P  could  be  located  for  corresponding  values  of  the  pri- 
mary current  and  that  the  locus  of  such  point  would  form  a 
continuous  curve  representing  the  value  and  time-phase  position 
of  the  primary  and  secondary  currents  throughout  the  operating 
range  of  the  motor. 

On  account  of  the  fact  that,  independent  of  the  method  by 
which  the  e.m.f.  may  be  produced  in  the  secondary  circuit  of 
the  motor,  the  primary  circuit  is  that  of  a  static  transformer, 
the  locus  of  the  primary  current  as  here  designated  approximates 
closely  a  circle,  the  center  of  which  is  located  on  the  line  B  A 
prolonged  and  the  diameter  of  which  is  equal  to  the  ratio  of  the 
impressed  e.m.f.  to  the  combined  local  leakage  reactance  of  the 
two  coils,  as  is  true  with  any  stationary  transformer.  The  posi- 
tion of  the  complete  circle  is  determined  and  it  may  readily  be 
drawn  both  to  the  right  and  to  the  left  of  the  origin  of  vectors, 
O  when  the  initial  point,  A,  and  any  load  point,  P,  are  located, 
as  was  discussed  in  the  preceding  chapter. 

From  the  current  locus  of  Fig.  38,  the  components  of  any 
chosen  value  of  primary  current  and  the  corresponding  secondary 
current  may  be  ascertained  at  once  and  when  the  resistances  of 
the  primary  and  secondary  coils  are  known,  the  complete  per- 
formance of  the  machine  may  be  determined  by  simple  calcula- 
tions. Such  calculations  are  recorded  in  the  table,  and  the 
results  thereof  are  represented  graphically  in  Fig.  39.  The 
methods  employed  in  obtaining  the  results  sought  will  be  ap- 
preciated from  a  study  of  the  head-lines  of  the  several  columns 
of  the  table. 

In  all  cases,  the  current  referred  to  is  the  "  equivalent  single- 
phase  current  "  or  the  corresponding  component  thereof.  This 
term  is  used  to  express  that  value  of  current  which  multiplied 


78 


ALTERNATING  CURRENT  MOTORS. 


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INDUCTION  MOTORS  AS  GENERATORS. 


70 


by  the  circuit  e.m.f.  and  power-factor  will  give  the  true 
watts;  and  the  use  of  such  term  greatly  facilitates  any  calcula- 
tions with  polyphase  quantities,  since  all  quantities  may  then 
be  treated  as  though  of  single-phase  significance. 

Similarly,  the  resistance  used  is  the  equivalent  single-phase 
quantity,  having  that  value  which  multiplied  by  the  square  of 
the  equivalent  single-phase  current  will  give  the  true  copper 
loss  of  the  circuit  considered.  It  is  an  interesting  fact,  previ- 
ously verified,  that,  for  any  given  two-  or  three-phase  circuit,  how- 
soever inter-connected,  the  equivalent  single-phase  resistance  is 


85  86       88        90        92       94 

Speed  in  Percent  of  Synchronism 


100      102      104      106      108      110      112      114  115  j 
Speed  in  Percent  otSynchronism  -* 


FIG.  39.— Test  of  Asynchronous  Machine  as  Motor  and  as 
Generator. 


just  one-half  of  the  value  found  between  leads  by  means  of  direct 
current  measuring  instruments. 

PERFORMANCE  CHARACTERISTICS. 

The  curves  of  Fig.  39  show  the  performance  of  the  asynchron- 
ous machine  when  its  speed  is  varied  from  85  per  cent,  con- 
tinuously to  115  per  cent,  of  synchronism  and  are  the  charac- 
teristic curves  which  would  be  experimentally  obtained  by 
belting  the  asynchronous  induction  machine  to  a  shunt-wound, 
direct-current  machine  which  being  supplied  with  constant 
e.m.f.  could  be  driven  throughout  this  range  of  speed  by  varia- 


80  ALTERNATING  CURRENT  MOTORS. 

tion  of  its  field  strength,  thereby  converting  it  from  a  generator 
to  a  motor  gradually,  as  desired. 

As  seen  from  Fig.  39,  at  synchronous  speed  the  induction 
motor  receives  all  of  its  power  electrically  from  the  supply 
system  and  delivers  no  mechanical  power.  At  106  per  cent,  of 
synchronism  this  particular  machine  receives  all  of  its  power 
mechanically  and  delivers  no  electrical  power  whatever.  Be- 
tween these  two  speeds,  all  power  received  by  the  machine  is 
dissipated  in  internal  losses  and  the  demand  for  power  by  the 
machine  is  gradually,  with  the  negative  slip,  transferred  from 
the  electrical  to  the  mechanical  source  of  supply.  Below  syn- 
chronism, the  machine  gives  out  mechanical  power,  while  above 
106  per  cent,  of  synchronism  the  power  given  out  is  electrical; 
that  is,  the  machine  operated  as  a  generator  and  returns  power 
to  the  electrical  supply  system. 

Such  an  asynchronous  generator  can  be  connected  directly  to 
the  supply  network  without  the  necessity  of  first  bringing  the 
machine  to  the  exact  speed  corresponding  to  the  circuit  fre- 
quency, and  the  portion  of  the  load  which  it  will  assume  can  be 
adjusted  quite  accurately  by  variation  of  the  speed  of  its  prime 
mover. 

PARALELL  OPERATION  OF  ASYNCHRONOUS  GENERATORS. 

An  asynchronous  generator,  as  here  assumed,  will  operate 
satisfactorily  in  parallel  with  generators  of  the  synchronous 
type — the  ordinary  alternators;  the  frequency  of  the  current 
delivered  by  it,  however,  is  in  any  case  less  than  that  corre- 
sponding to  the  speed  of  its  rotor,  the  difference  being  due  to 
the  requisite  motion  of  the  secondary  conductors  with  reference 
to  the  primary  field.  The  division  of  the  load  between  synchron- 
ous generators  working  in  parallel  is  determined  by  the  relative 
phase  position  of  the  machines;  the  generator  which  maintains 
a  phase  position  in  advance  of  others  carrying  the  greatest 
load,  while  one  which  lags  behind  in  phase  position  is  more 
lightly  loaded.  The  actual  speed  of  all  the  generators  thus  con- 
nected, however,  must  be  the  same;  it  is  merely  the  tendency 
to  different  speeds,  as  fixed  by  the  governing  mechanism  of 
the  prime  movers,  which  determines  the  load  division.  As  far 
as  concerns  the  prime  movers,  the  condition  of  parallel  operation 
of  asynchronous  generators  is  quite  similar  to  the  above,  but 


INDUCTION  MOTORS  AS  GENERATORS.  81 

in  the  latter  case  the  load  division  is  determined  by  the  actual 
operating  speed  of  each  individual  machine;  that  is,  the  load 
carried  by  each  is  determined  wholly  by  the  variation  of  the 
speed  of  its  rotor  from  that  corresponding  to  the  circuit  fre- 
quency. When  its  speed  is  'below  the  circuit  frequency,  an 
asynchronous  generator  is  driven  as  an  induction  motor;  which 
fact  allows  an  asynchronous  generator  to  be  connected  to  the 
operating  circuit  without  being  brought  to  exact  synchronous 
speed. 

EXCITATION  OF  ASYNCHRONOUS  GENERATORS. 

As  is  true  with  the  synchronous  alternator,  the  asynchronous 
generator  is  not  inherently  self-exciting  but  current  for  excita- 
tion must  be  supplied  from  some  source  external  to  itself. 
Similarly,  if  the  delivered  (or  supplied)  e.m.f.  is  to  be  kept  con- 
stant, the  exciting  current  must  be  increased  as  the  load  in- 
creases. When  the  asynchronous  generator  is  delivering  power 
to  constant  potential  mains,  the  exciting  current  automatically 
adjusts  itself  to  correspond  to  the  demands  of  the  load.  This 
characteristic  of  the  machine  will  be  appreciated  from  a  study 
of  the  curve  of  Fig.  39,  marked  "  wattless  current."  Such 
curve,  when  plotted  to  the  proper  coordinates,  resembles  closely 
the  curve  representing  the  relation  between  external  load  am- 
peres and  internal  field  current  of  the  synchronous  alternator. 

A  further  analysis  of  the  various  components  of  the  currents 
reveals  the  fact  that,  when  operating  as  either  a  motor  or  a 
generator,  the  asynchronous  machine  possesses  definite  load- 
speed  characteristics  which  are  unalterable  by  any  condition 
external  to  the  machine.  Thus,  at  any  certain  speed,  the 
machine  demands  from  the  network  a  definite  value  of  wattless 
current  and  delivers  a  certain  amount  of  power  either  electrical 
or  mechanical  independent  of  the  requirements  of  other  machines 
connected  to  the  system.  As  a  generator,  the  machine  can 
deliver  no  wattless  current  whatsoever  and  its  own  supply  of 
such  current  must  be  derived  from  some  other  source. 

When  an  asynchronous  generator  and  a  synchronous  motor 
or  converter  are  connected  simultaneously  to  the  supply  sys- 
tem, the  field  excitation  of  the  synchronous  machine  may  be  so 
adjusted  as  to  cause  this  machine  to  supply  the  amount  of 
wattless  current  demanded  by  the  asynchronous  generator, 


82 


ALTERNATING  CURRENT  MOTORS. 


while  the  speed  of  the  asynchronous  generator  may  be  so  regu- 
lated that  it  supplies  just  that  amount  of  power  current  required 
for  the  synchronous  machine.  Under  these  conditions,  the 
machines  so  interchange  currents  that  they,  considered  as  a 
unit,  may  be  disconnected  from  the  network  without  affecting 
the  operation  of  either  machine.  The  e.m.f.  of  the  set  may  be 
adjusted  as  desired  by  variation  of  the  field  strength  of  the 
synchronous  machine.  The  inherent  regulation  of  the  e.m.f. 
of  the  set,  as  the  load  thereon  is  varied,  depends  upon  the  mag- 
netic circuits  of  the  two  machines  and,  in  general,  the  per- 
formance is  stable  only  when  at  least  one  of  the  machines  is 
operated  under  conditions  appro ximating  magnetic  saturation. 
With  circuits  arranged  as  in  Fig.  40,  it  may  be  shown  experi- 
mentally that,  under  any  given  condition  of  operation,  all 


Synchronous  Motor 
or  Converter 


Asynchronous 
Generator 

FIG.  40. — Arrangement  of  Load  and  Exciter  Circuits;  Syn- 
chronous Converter  Excitation. 


wattless  current  comes  from  the  synchronous  machine  while 
all  power  current  comes  from  the  asynchronous  generator. 
The  synchronous  converter  may  supply  power  electrically  from 
its  commutator,  mechanically  from  its  shaft,  or  power  may  be 
derived  directly  from  the  mains  at  the  generator.  The  slip  of 
the  rotor  from  the  speed  corresponding  to  the  circuit  frequency 
will  depend  upon  the  amount  of  power  thus  demanded,  so  that 
with  constant  rotor  speed,  the  circuit  frequency  will  vary  with 
the  load  on  the  set.  The  speed  of  the  synchronous  machine 
will  vary  with  the  circuit  frequency,  so  that,  independent  of 
the  effect  of  any  wattless  current  which  may  be  demanded, 
operation  at  constant  circuit  e.m.f.  can  be  obtained  only  when 
the  field  strength  of  the  synchronous  machine  is  varied  with 
the  load. 


INDUCTION  MOTORS  AS  GENERATORS.  s:{ 

Any  demand  for  lagging  current  from  the  set  necessitates  an 
increase  in  the  lagging  component  of  current  from  the  syn- 
chronous motor  and  weakens  the  field  strength  of  this  machine 
and  thereby  lowers  the  circuit  e.m.f.  A  demand  for  leading 
current  produces  the  opposite  effect,  and,  if  the  leading  current 
be  properly  adjusted  in  value,  no  wattless  current  whatever 
will  flow  from  the  synchronous  machine  and  it  may  be  discon- 
nected from  the  circuit  without  affecting  the  performance  of 
the  asynchronous  generator.  An  additional  synchronous  motor 
with  its  field  cores  over-excited  may  be  used  as  a  source  of  leading 
current,  or  static  condensance  with  proper  means  for  adjust- 
ment may  be  employed  for  this  purpose. 

CONDENSERS  AS  A  SOURCE  OF  EXCITING  CURRENT. 

If  the  source  of  excitation  be  a  synchronous  alternator,  the 
effect  of  the  presence  of  the  exciting  current  for  the  asynchronous 


FIG.  41. — Asynchronous  Generator;  Condenser  Excitation. 

machine  may  be  eliminated  by  placing  across  the  generator 
circuit  condensers  adjusted  in  capacity  so  as  to  take  an  amount 
of  leading  current  equal  to  the  lagging  current  of  the  asyn- 
chronous generator.  (See  Fig.  41.)  If  this  adjustment  be 
exact,  and  there  be  no  fluctuations  in  the  load,  no  current  will 
pass  from  the  synchronous  to  the  asynchronous  machine,  and 
the  circuit  between  them  may  be  opened  without  producing  any 
effect  whatever  upon  tne  operation  of  either  machine.  Under 
the  conditions  here  assumed  the  asynchronous  generator  will 
receive  its  exciting  current  from  the  condensers,  and,  so  long 
as  the  load  remains  constant  and  no  changes  occur  in  the  speed 
of  the  prime  mover,  it  will  automatically  maintain  its  e.m.f. 
at  a  constant  value.  If  additional  load  be  now  placed  upon 
the  generator,  the  e.m.f.  will  decrease,  while,  if  the  load  be 
decreased  or  removed  entirelv,  the  e.m.f.  will  at  once  increase 


84  ALTERNATING  CURRENT  MOTORS. 

the  performance  being  similar  in  many  respects  to  that  of  a 
direct-current  shunt-wound  generator. 

It  is  believed  that  the  condenser  excitation  of  asynchronous 
generators  offers  sufficiently  varied  application  of  the  charac- 
teristics of  the  apparatus  involved  to  justify  an  extended  dis- 
cussion of  its  use. 

Before  passing  to  a  discussion  of  the  performance  of  an 
asynchronous  generator  when  excited  by  static  condensance 
it  may  be  well  to  recall  a  few  of  the  fundamental  facts  con- 
nected with  the  operation  of  condensers  in  alternating  current 
circuits. 

CONDENSERS  IN  ALTERNATING-CURRENT  CIRCUITS. 

When  two  conducting  bodies  in  close  proximity  are  connected 
to  opposite  terminals  from  a  source  of  electrical  energy,  it  is 
found  that  there  accumulates  upon  the  adjacent  surfaces  or 
within  the  intervening  insulating  material,  i.e.,  the  dielectric, 
a  condition  of  sub-atomic  activity  termed  "  electricity."  Other 
conditions  remaining  the  same,  the  amount  of  electricity,  or 
the  charge  which  the  bodies  will  store  under  a  certain  potential 
difference  varies  inversely  as  the  distance  separating  the  plates, 
and  is  greatly  affected  by  the  character  of  the  dielectric.  An 
assembly  of  numerous  conducting  plates  separated  by  sheets 
of  dielectric  material  as  thin  as  practicable  and  yet  of  sufficient 
thickness  to  give  the  requisite  insulating  strength  to  withstand 
the  maximum  e.m.f.  to  be  impressed  at  the  terminals,  forms  the 
essential  features  of  the  commercial  condenser. 

In  order  now  to  investigate  the  action  of  a  given  condenser, 
assume  one  connected  to  a  source  of  electrical  energy,  of  which 
the  e.m.f.  may  be  changed  at  will,  Assume  further  that  at  the 
moment  of  connecting  the  condenser  in  circuit,  the  e.m.f.  is 
of  zero  value  but  increasing  in  a  positive  direction;  charge  will 
pass  into  the  condenser,  tending  to  produce,  by  strain  in  the 
dielectric,  a  counter  e.m.f.  equal  to  the  impressed.  Evidently 
so  long  as  the  e.m.f.  continues  to  increase,  charge  will  continue 
to  pass  into  the  condenser  at  a  rate  proportional  to  the  in- 
stantaneous increase  in  e.m.f.  When  the  e.m.f.  reaches  its 
maximum  value,  the  condenser  will  have  received  its  full  charge 
and  no  additional  amount  of  electricity  will  flow  thereto.  As 
the  e.m.f.  decreases,  charge  will  flow  from  the  condenser  tending 


INDUCTION  MOTORS  AS  GENERATORS.  85 

at  each  instant  to  make  the  amount  remaining  therein  correspond 
to  the  instantaneous  value  of  e.m.f.  From  the  above  facts  it 
will  be  seen  that  the  rate  of  transfer  of  charge,  i.e.,  the  current 
in  the  circuit,  will  have  a  value  represented  by  the  rate  of 
change  of  the  e.m.f.  and  if  the  e.m.f.  follows  a  sine  curve  of 
time-value  the  current  will  follow  the  corresponding  cosine  curve, 
and,  as  seen  above,  will  be  90  times  degrees  ahead  of  the  e.m.f. 

For  purpose  of  comparison  of  the  performance  of  different 
condensers,  it  is  convenient  to  specify  the  amount  of  charge 
which  a  given  condenser  will  assume  under  a  certain  e.m.f. 
relative  to  some  charge  taken  as  a  unit.  If  a  certain  con- 
denser, when  subjected  to  a  continuous  pressure  of  one  volt, 
accumulates  electricity  to  the  amount  of  one  coulomb,  that  is 
the  amount  of  electricity  represented  by  one  ampere  for  one 
second,  it  is  said  to  possess  a  capacity  of  one  farad.  A  con- 
denser of  capacity  C  is  one  which  under  a  continuous  electro- 
motive force  of  one  volt  assumes  a  charge  of  C  coulombs  or 
when  the  e.m.f.  is  E  volts  the  charge  will  be  E  C  coulombs. 

When  the  e.m.f.  changes  there  is  a  flow  of  current  to  or  from 
the  condenser  so  that  at  each  instant  the  charge  in  the  con- 
denser tends  to  adjust  itself  to  correspond  with  the  instanta- 
neous value  of  the  e.m.f.  If  the  e.m.f.  changes  at  a  constant 
rate  of  one  volt  per  second,  then  the  charge  must  vary  C  coulombs 
per  second,  or  the  current  must  have  a  steady  value  of  C  am- 
peres, or,  in  general,  the  current  must  be  C  times  the  rate  of 
change  of  the  e.m.f.,  that  is 


now  e  =  Em  sin  at  t 

where       Em  is  maximum  e.m.f.  ^and  aj,   electrical 
angular  velocity, 

de 

hence  —r  =  a>  Em  Cos.  co  t  so  that 
at 

i  =-  '  C  to  Em  Cos.  to  t  or  the  maximum  current,  7m  =  C  co  Em, 
and,  virtual  Values  being  used  throughout, 

7  =  CwE 
which  means  that  if  a  condenser  of  capacity  C  be  connected  to 


86  ALTERNATING  CURRENT  MOTORS. 

a  source  of  alternating  e.m.f.  E  when  the  frequency  is  /  there 
will  flow  therein  an  alternating  current  of  value  /  such  that 

/  =  c  Ew  =  C  £  2  TT  /. 

In  alternating  current  circuits,   the  ratio  E  to  I  gives  the 
impedance  which  for  a  condenser  as  above  is 

E_          E          J^_ 

"~ 


to  which  specific  quantity  there  is  given  the  name  condensance, 
or  negative  reactance. 

In  the  statement  above  concerning  the  angle  of  lead  of  the 
current  taken  by  a  condenser,  a  fact  of  minor  importance  has 
been  neglected.  It  is  found  that  the  value  of  energy  which  a 
condenser  returns  upon  discharge  is  less  than  that  received 
during  charge,  the  difference  being  dissipated  in  heat  loss  within 
the  condenser  through  dielectric  hysteresis  and  to  a  small 
extent  to  the  heating  effect  of  the  current  upon  the  conducting 
material.  This  loss  requires  a  component  of  current  in  phase 
with  the  e.m.f.  across  the  condenser  terminals,  so  that  in  any 
practical  condenser  the  current  leads  the  applied  e.m.f.  by  a 
time-angle  less  than  90°.  The  energy  component  of  the  con- 
denser current  is  relatively  quite  small  and  for  most  practical 
purposes  may  be  neglected. 

Having  taken  this  glance  at  the  inherent  characteristics  of 
condensers,  we  are  prepared  to  investigate  the  effect  of  con- 
necting such  condensers  to  the  operating  circuits  of  an  asyn- 
chronous generator. 

EXCITATION  CHARACTERISTICS  OF  ASYNCHRONOUS  GENERATORS. 

Lines  OL,  OM  and  ON  of  Fig.  42  show  the  relation  existing 
between  the  e.m.f.  impressed  upon  certain  condensers  L,  M 
and  N  and  the  current  taken  by  them  at  constant  frequency. 
As  will  be  noted  from  the  equations  developed  above,  the  effect 
of  each  condenser  circuit  can  be  represented  by  a  right  line 
passing  through  the  origin  and  the  slope  of  the  line  depends 
directly  upon  the  amount  of  effective  condensance  in  the  circuit 
considered.  The  line  OPRTV  is  the  excitation  characteristic 
of  an  asynchronous  generator  when  driven  at  constant  speed 
without  load  and  is  found  as  the  relation  of  the  impressed  e.m.f. 


INDUCTION  MOTORS  AS  GENERATORS. 


87 


to  the  wattless,  lagging,  component  of  the  current  in  the  primary 
coil. 

If  condensance  of  some  value,  such  as  M,  be  connected  to  the 
supply  system  in  parallel  with  the  asynchronous  generator,  then, 
at  any  impressed  e.m.f.,  the  condensance  will  require  a  definite 
amount  of  leading  current  while  the  lagging  current  of  the 


0        10      20      30      40      50      60       70      80      90 
Wattless  Component  of  Current 

FIG.  42. — Characteristics  of  Condensers  and 
Excitation  Characteristics  of  Three-phase 


100    110 


Synchronous 
Machine. 


asynchronous  generator  will  likewise  be  of  a  definite  amount. 
With  the  impressed  e.m.f.  properly  adjusted,  the  leading  current 
taken  by  the  condensance  will  equal  the  lagging  current  of 
the  generator  and  no  wattless  current  will  be  supplied  from 
any  other  source.  Such  value  of  impressed  e.m.f.  is  found  in 
Fig.  42  at  the  intersection  of  the  condenser  line  O  M  with  the 
generator  excitation  characteristics  at  T,  the  value  here  being 


88  ALTERNATING  CURRENT  MOTORS. 

approximately  292  volts.  With  conditions  existing  as  here 
assumed,  the  asynchronous  generator  and  condenser  exciter 
may  be  isolated  from  the  supply  system  and  the  set  will  auto- 
matically maintain  its  e.m.f.  at  292  volts. 

If  at  the  moment  of  disconnecting  the  asynchronous  generator 
and  the  condensance  M,  as  a  unit,  from  the  supply  system,  the 
impressed  e.m.f.  be  220  volts, — the  normal  value  for  the  asyn- 
chronous machine, — the  e.m.f.  of  the  set  will  at  once  increase 
to  292  volts,  due  to  the  following  causes:  Under  220  volts  the 
condensance  takes  42  amperes  leading  current,  which,  when 
supplied  from  the  asynchronous  machine  would  raise  its  e.m.f. 
to  256  volts,  at  which  e.m.f.  the  condensance  would  take  49 
amperes,  raising  the  e.m.f.  of  the  generator  to  276  volts,  and  so 
on,  until  the  e.m.f.  became  stable  at  292  volts,  as  shown  at  Tin 
Fig.  42.  If,  now,  the  condensance  in  circuit  be  changed  to  some 
value  such  as  is  represented  by  the  line  ON,  the  e.m.f.  will  at 
once  rise  to  the  value  shown  at  the  intersecting  point,  V.  on 
the  excitation  characteristics.  With  a  value  of  condensance 
such  as  OL  in  circuit,  the  generator  will  be  unable  to  maintain 
its  excitation  at  any  e.m.f.  whatsoever,  since  there  is  no  point 
of  intersection  with  the  excitation  characteristic. 

Due  to  the  extremely  weak  magnetic  condition  in  which  a 
ring  core  without  projecting  poles  is  left  when  the  exciting 
force  is  removed,  and  also  to  a  large  extent  to  the  lower  initial 
inverted  knee  of  the  excitation  characteristic  which  requires  a 
relatively  large  exciting  force  in  comparison  with  the  e.m.f. 
produced  at  this  point,  the  generator  possesses  but  slight  ten- 
dency to  build  up  from  its  remnant  magnetism  when  con- 
densance of  normal  operating  value  is  connected  in  circuit. 
It  will  be  recalled  that  such  behavior  is  characteristic  also  of 
shunt-wound,  direct-current  generators.  In  fact,  for  a  given 
resistance  in  the  shunt  coil  circuit,  the  relation  between  the 
current  flowing  therein  and  the  e.m.f.  is  a  right  line  similar  to 
the  condensance  lines  in  Fig.  42  and  the  slope  of  such  line  deter- 
mine the  e.m.f.  up  to  which  the  machine  will  build.  That  the 
shunt  circuit  resistance  must  ordinarily  be  decreased  below  the 
operating  value  before  the  direct-current  generator  will  build 
up  from  its  residual  magnetism  is  familiar  to  all.  This  is  due 
to  the  fact  that  the  slope  of  the  shunt  circuit  current  line  is 
such  as  to  cause  it  to  intersect  with  the  excitation  characteristic 


INDUCTION  MOTORS  AS  GENERATORS. 


89 


at  the  lower  inverted  knee  of  the  initial  line  of  the  hysteresis  loop. 

With  the  asynchronous  generator,  current  of  any  frequency 
and  of  almost  any  value,  or  a  static  charge  in  the  condensers 
may  be  used  to  cause  the  machine  to  build  up.  If  the  con- 
densance,  when  connected  in  the  circuit,  is  above  a  certain 
value  depending  upon  the  magnetic  circuit  of  the  machine,  the 
generator  will  build  up  instantly  from  its  remnant  magnetism 
without  initial  external  excitation. 

Fig.  43  shows  the  connecting  circuits  for  condenser  excitation 
of  an  asynchronous  generator  and  indicates  a  convenient  method 
for  varying  the  amount  of  effective  condensance  in  circuit. 
For  sake  of  simplicity,  the  diagram  is  made  to  represent  single- 


— <&= 


<s>— I 


Condenser 
Exciter 


Asynchronous 
Genjanator 

FIG.  43. — Arrangement  of  Load  and  Exciter  Circuits;  Con- 
denser Excitation. 


phase     circuits,     although     polyphase     equipment     throughout 
could  similarly  be  employed. 

By  the  use  of  a  transformer  or  auto  transformer,  the  effective 
condensance  in  circuit  can  be  adjusted  within  range,  to  any  value 
desired  by  variation  of  the  ratio  of  turns  of  the  transformer 
coils,  without  in  any  manner  changing  the  actual  condensance 
connected  thereto;  the  effective  condensance  varying  as  the 
square  of  the  ratio  of  turns.  This  relation  will  be  appreciated 
when  it  is  considered  that  if,  when  the  ratio  is  1  to  1,  the  con- 
densance takes  current  /,  when  the  ratio  is  increased  to  1  to  M, 
the  primary  e.m.f.  remaining  the  same  as  before,  the  secondary 
condensance  current  will  be  M I  and  the  primary  opposing 
current  will  be  M  times  as  great  or  M2  /.  It  is  thus  seen  that, 
although  the  source  of  excitation  of  the  generator  is  the  con- 


90  ALTERNATING  CURRENT  MOTORS. 

densers,  an  increase  in  the  step  down  ratio  of  transformation  of 
the  e.m.f.  from  the  condensers  will  result  in  an  increase  in  the 
generated  e.m.f. 

LOAD  CHARACTERISTICS  OF  ASYNCHRONOUS  GENERATORS. 

The  load  characteristics  of  an  asynchronous  generator,  when 
excited  by  static  condensers,  are  quite  similar  to  those  of  the 
familiar  shunt-wound,  direct  current  machine;  an  increase  in 
load  producing  a  decrease  in  e.m.f.,  due  both  to  the  direct  effect 
of  the  load  on  the  armature  of  the  machine  and  to  the  added  effect 
of  the  decrease  in  exciting  current  from  the  lessened  e.m.f.  at 
the  field  circuit.  With  the  asynchronous  machine  a  further 
effect  is  produced  by  the  change  in  circuit  frequency  with  the 
load,  when  the  rotor  is  driven  at  constant  speed.  Since  the 
effective  condensance  for  any  given  adjustment  varies  directly 
with  the  frequency,  and  the  exciting  current  to  produce  a 
certain  e.m.f.  for  the  asynchronous  machine  varies  inversely 
therewith,  all  other  conditions  remaining  the  same,  a  mere 
change  in  frequency  will  alter  the  e.m.f.  at  which  the  set  will 
operate.  Since  even  a  non-inductive  resistance  drop  of  e.m.f. 
in  the  generator  windings  under  load  current  causes  a  slight 
decrease  of  the  current  in  the  exciter  circuit  under  the  lessened 
terminal  e.m.f.,  the  best  regulation  is  obtained  when  the  con- 
denser is  connected  across  one  set  of  windings  and  the  load 
placed  across  an  independent  set,  as  shown  in  Fig.  44. 

With  the  familiar  synchronous  alternator  a  lagging  load 
current  tends  to  decrease  the  terminal  e.m.f.  while  a  leading 
current  has  the  opposite  effect.  An  exactly  similar  state  of 
affairs  exists  with  a  self-excited  asynchronous  generator.  In 
this  case,  however,  the  effect  is  cumulative,  since  any  change 
in  the  terminal  e.m.f.  causes  a  variation  of  the  current  in  the 
condenser  exciter  and  the  field  magnetism  must  adjust  itself 
to  a  correspondingly  altered  valve. 

If  it  were  possible  always  to  operate  the  set  at  constant 
frequency,  then,  by  use  of  Figs.  39  and  42,  one  could  de- 
termine at  once  the  amount  of  condensance  necessary  to  main- 
tain the  e.m.f.  constant  for  any  load,  and  conversely  the  in- 
herent regulation  of  the  set  could  be  ascertained.  Take,  for 
example,  the  condition  of  operation  represented  in  Fig.  39, 
at  105.5  per  cent,  speed.  The  machine  is  delivering  16.5  horse- 


INDUCTION  MOTORS  AS  GENERATORS. 


91 


power  at  an  efficiency  of  80  per  cent.,  and  requires  42  amperes 
when  the  e.m.f.  is  220  volts.  The  condensance  necessary  to  give 
42  amperes  leading  current  at  220  volts  is  shown  in  Fig.  42 
by  the  line  O  M.  If  the  load  be  removed  from  the  set  and 
the  circuit  frequency  remain  constant,  that  is,  if  the  rotor 
speed  be  decreased  to  100  per  cent.,  the  e.m.f.  will  increase  to 
292  volts  as  indicated  by  point  T  in  Fig.  42.  If,  however,  the 
rotor  speed  remained  at  105.5  per  cent,  or  increased  when  the 
load  was  removed,  the  e.m.f.  would  reach  a  much  higher  value. 
It  is  thus  seen  that  close  regulation  necessitates  that  the  ma- 
chine be  operated  above  the  knee  of  the  excitation  character- 


condenser 


Condense 


FIG.  44. — Exciter  Circuits  for  Asynchronous  Generator. 

istic  (Fig.  42)  and  that  the  slip  under  load  be  small,  that  is, 
that  the  secondary  resistance  be  small,  as  has  been  mentioned 
previously. 

Since  magnetic  saturation  of  material  subjected  to  flux  alter- 
nating at  high  frequency  means  excessive  iron  loss,  efficiency  of 
performance  dictates  that  the  core  of  an  asynchronous  generator 
be  so  designed  that  the  secondary  member,  which  is  subjected 
to  the  low  frequency  of  reversal  corresponding  to  the  slip 
reaches  the  saturation  point  while  the  primary  member  is  yet 
much  below  such  condition. 

When  it  is  remembered  that  the  amount  of  current  taken  by 
a  condenser  in  an  alternating-current  circuit  under  a  certain 


92  ALTERNATING  CURRENT  MOTORS. 

e.m.f.,  varies  directly  with  the  frequency,  and  that  the  number 
of  magnetic  lines  of  force  to  produce  a  given  e.m.f.  in  the  gen- 
erator coils  varies  inversely  with  the  frequency,  it  will  be 
appreciated  that  the  e.m.f.  which  a  certain  condenser  will  give 
to  an  asynchronous  generator  will  depend  largely  upon  the  fre- 
quency. The  frequency  is  determined  primarily  by  the  speed 
of  the  rotor,  but  it  decreases,  even  for  a  constant  rotor  speed, 
when  the  generator  load  increases;  a  fact  which  shows  the 
importance  of  good  speed  regulation  at  the  prime-mover.  Any 
leading  current  taken  by  the  load  acts  as  additional  condenser 
capacity  to  increase  the  generated  e.m.f.,  while  lagging  current 
produces  the  opposite  effect;  in  fact,  a  condenser-excited  gen- 
erator of  this  type  which  operates  satisfactorily  under  a  non- 
inductive  load,  may  be  caused  to  lose  its  e.m.f.  entirely  by  the 
addition  of  a  relatively  small  proportion  of  inductive  load. 

COMMUTATOR  EXCITATION  OF  ASYNCHRONOUS  GENERATORS. 

Mr.  Heyland  has  devised  a  method  for  causing  the  asyn- 
chronous generator  to  supply  its  own  current  for  excitation. 
In  the  application  of  his  method,  current  is  taken  from  the 
main  circuit  of  the  generator  and  passed  to  the  secondary 
conductors  through  a  suitable  commutator  on  the  rotor.  The 
action  of  the  commutator  in  supplying  the  exciting  current  will 
be  appreciated  by  first  considering  two  extreme  conditions  of 
operation.  If  direct  current  be  introduced  by  way  of  slip  rings 
into  the  secondary  windings  of  an  asynchronous  generator 
while  the  rotor  is  driven  at  normal  synchronous  speed,  it  will 
be  found  that  alternating  current  at  normal  frequency  may 
be  obtained  from  the  primary  windings  and  that  the  e.m.f. 
generated  may  be  adjusted  in  value  by  a  corresponding  change 
in  the  direct  current  supplied.  In  fact,  one  readily  appreciates 
that  operating  under  the  conditions  here  assumed  the  asyn- 
chronous machine  is  converted  into  a  simple  alternating-cur- 
rent generator  and  possesses  all  of  the  characteristics  of  this 
type  of  machine. 

If  with  the  rotor  stationary  alternating  current  of  normal 
frequency  be  supplied  to  the  secondary  windings,  current  at 
the  same  frequency  may  be  derived  from  the  primary  windings. 
Under  these  conditions,  the  generator  is  again  a  source  of  alter- 
nating current,  but  it  now  possesses  the  inherent  characteristics 


INDUCTION  MOTORS  AS  GENERATORS. 


93 


of  a  stationary  transformer.  If  now  a  commutator  be  con- 
nected to  the  secondary  windings,  any  motion  of  the  rotor  in 
either  direction  will  not  alter  the  effect  of  any  certain  value  of 
current  in  the  secondary  in  producing  e.m.f.  in  the  primary 
winding,  since  the  space-phase  position  of  the  current  with  refer- 
ence to  the  primary  coils  will  be  the  same  as  would  be  the  case 
were  the  rotor  stationary.  Hence,  independent  of  the  speed 
of  the  rotor,  the  current  thus  introduced  into  the  secondary 
reacts  upon  the  primary  with  the  primary  frequency.  The 
value  of  the  current  in  the  secondary  can  be  varied  by  changing 
the  e.m.f.  impressed  upon  the  commutator,  and  it  may  be  given 
any  phase  position  with  reference  to  its  reaction  upon  the 


Secondary  with  Short- 
circuited  Commutator 
Compound  Excitation 

FIG.  45. — Asynchronous  Generator;  Commutator  Excitation. 


primary  by  changing  the  position  of  the  rotor  brushes  relative 
to  the  field  coils.  (See  Fig.  45.) 

When  the  rotor  is  driven  at  synchronous  speed  in  the  direc- 
tion of  the  revolving  field,  the  e.m.f.  required  to  be  impressed 
upon  the  secondary  windings  in  order  to  cause  a  given  current 
to  flow  there  through  is  greatly  reduced  below  the  value  neces- 
sary when  the  rotor  is  stationary,  due  to  the  practical  elimina- 
tion of  the  reactive  e.m.f.  at  the  low  frequency  of  the  current 
in  the  individual  slots  containing  the  secondary  conductors. 

The  operation  of  an  asynchronous  generator  with  commutator 
excitation  is  quite  similar  to  that  of  one  excited  by  means  of 
condensers,  and  the  statements  made  above  as  to  the  effect 
of  speed  variations  and  of  the  character  of  the  load  current 


94  ALTERNATING  CURRENT  MOTORS. 

upon  the  delivered  e.m.f.,  and  as  to  the  necessity  of  working 
the  magnetic  core  at  high  density  apply  equally  to  the  Heyland 
shunt-excited  asynchronous  generator.  With  a  commutator 
generator,  however,  it  is  possible  to  use  compound  excitation, 
which  consists  in  passing  the  load  current,  by  means  of  an 
auxiliary  set  of  brushes  on  the  commutator,  through  the  sec- 
ondary conductors,  and  thus  to  increase  the  field  strength 
with  the  load  and  to  counteract  the  effect  of  any  lagging  current 
upon  the  field  magnetism  and  the  circuit  e.m.f.  This  latter 
action  is  very  similar  to  that  produced  in  direct-current  gen- 
erators equipped  with  the  Ryan  balancing  coils.  Since  a  load 
current  which  lags  in  the  primary  coils  will  lag  equally  in  the 
auxiliary  exciting  circuit  of  the  secondary,  it  is  possible  by 
means  of  the  compound  excitation  to  compensate  for  the  field 
demagnetizing  effect  of  an  inductive  load  upon  the  generator. 


CHAPTER  IX. 
TRANSFORMER  FEATURES  OF  THE  INDUCTION  MOTOR. 

ELECTRIC  AND  MAGNETIC  CIRCUITS. 

Fig.  46  represents  the  electric  and  magnetic  circuits  of  an  ideal 
transformer.  When  an  alternating  e.m.f.  is  impressed  upon  the 
primary  terminals,  the  secondary  being  on  open  circuit,  a  certain 
value  of  current  flows  in  the  coil.  The  magnetomotive  force 
due  to  the  ampere -turns  of  this  current  causes  lines  of  force  to  be 
produced  in  the  core,  and  the  change  in  the  value  of  these  lines  with 
the  alternation  of  the  current  generates  in  the  primary  coil  an 
e.m.f.  in  a  time-phase  position  to  tend  to  decrease  the  current 
in  the  coil;  the  final  result  being  that  there  flows  in  the  coil, 


0  o  0 


FIG.  46. — Electric  and  Magnetic  Circuits  of  an  Ideal  Transformer. 


just  that  value  of  current  whose  product  with  the  number  of 
primary  turns  gives  the  magnetomotive  force  necessary  to  send 
through  the  reluctance  of  their  path  that  number  of  lines 
the  change  in  the  value  of  which  generates  in  the  primary  coil 
an  e.m.f.  less  than  the  impressed  by  an  amount  just  sufficient 
to  allow  this  value  of  current  to  flow  through  the  local  im- 
pedance of  the  primary  coil.  If  the  local  impedance  of  the 
primary  coil — which  is  composed  of  the  resistance  and  the 
local  reactance  due  to  the  leakage  lines  surrounding  only  this 
coil — be  of  small  value,  the  e.m.f.  counter  generated  in  the 

95 


96  ALTERNATING  CURRENT  MOTORS. 

coil  by  the  alternating  flux  in  the  core  will  be  practically  equal 
to  the  impressed  e.m.f. 

The  effect  of  varying  the  reluctance  of  the  magnetic  path  in 
the  core  is  to  vary  accordingly  the  exciting  magnetomotive 
force,  but  no  appreciable  effect  is  produced  upon  the  value  of 
the  flux.  A  negligible  effect  may  be  attributed  to  the  changed 
value  of  exciting  current  through  the  slightly  varied  local  re- 
actance and  the  resistance  of  the  primary  coil,  which  may 
alter  the  diminutive  loss  of  e.m.f.  through  this  impedance. 
This  effect,  which  throughout  the  operating  range  of  well  de- 
signed transformers  is  augmented  to  a  negligible  extent  by 
the  load  current,  will  be  treated  more  in  detail  later. 

The  secondary  coil  is  placed  mechanically  in  a  position  to  be 
cut  by  the  greatest  proportion  of  the  flux  due  to  the  primary 
exciting  magnetomotive  force.  With  this  coil  on  open  circuit, 
there  will  be  generated  in  each  of  its  turns  by  the  change  in  the 
va'ue  of  core  flux,  an  e.m.f.  equal  to  the  counter  e.m.f.  per  turn  in 
the  primary.  If  the  secondary  circuit  be  closed  through  an  im- 
pedance, current  will  flow,  due  to  the  secondary  e.m.f.,  and  this 
current  will  tend  to  decrease  the  core  flux.  The  e.m.f.  counter 
generated  in  the  primary  being  somewhat  lessened,  more  cur- 
rent will  flow  therein  tending  to  restore  the  flux  to  its  former 
value,  and  stable  conditions  will  be  reached  when  the  additional 
primary  current  has  a  value  and  phase  position  such  as  to  give 
the  magnetomotive  force  necessary  to  counterbalance  the  effect 
of  the  secondary  ampere  turns,  thus  keeping  the  flux  in  the  core 
quite  closely  constant  at  the  value  demanded  by  the  primary 
e.m.f. 

The  exact  relation  between  the  flux  in  the  core,  the  frequency 
of  the  supply  current,  the  primary  counter  e.m.f.  and  number 
of  turns  can  be  derived  quite  simply  from  that  law  of  physics 
which  states  that  one  c.g.s.  unit  of  e.m.f.  is  generated  when 
flux  cuts  a  conductor  at  the  rate  of  one  line  per  second.  In 
general 

d<f> 


e  = 


dt 


Assuming  the  flux  (and  the  e.m.f.)  to  vary  with  time  in  a 
manner  to  be  represented  by  the  familiar  sine  law,  its  value  can 
be  stated  as 

<p  =  A  Bm  sin  a>  t 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.      97 

where  A  Bm  is  the  maximum  value  of  the  total  flux  over  the  area 
of  the  core  A. 

Thus  there  is  obtained 

d  (A  Bm  sin  aj  t) 

-AT 

e  =  A  Bm  a)  cos  <jj  t 

which  becomes  maximum  when  cos  a>  t  =  1  or 

77     _   A  D 
n,m  —  /i  nm  at 

so'  that  the  virtual  value  of  the  e.m.f.  per  turn  expressed  in 
c.g.s.  units  is 

A    Bm    OJ 


which  for  TV  turns  when  expressed  in  volts  becomes, 

ABm2xfN          .,ABmfN 
v/2  10s  ~W 

from  which  the  total  magnetic  flux,  A  Bm,  or  the  flux  density, 
Bm,  may  be  determined. 

Due  to  the  internal  friction  in  turning  the  molecular  magnets 
in  first  one  and  then  the  other  direction  with  the  alternation  of 
the  core  flux,  there  is  dissipated  a  certain  amount  of  energy 
with  each  reversal  of  flux,  such  energy  appearing  as  heat  in  the 
core  material  and  requiring  in  the  primary  coil  a  certain  com- 
ponent of  current  in  phase  with  the  impressed  e.m.f.  to  supply 
this  loss.  The  watts  thus  required  vary  with  the  1.6  power 
of  the  flux  density  and  with  the  quality  of  the  magnetic  mate- 
rial, and  can  be  expressed  thus, 


107 

where  Bm  =  maximum  flux  in  lines  per  sq.  c.m. 
V  =  volume  of  core  in  cubic  c.m. 
/  =  frequency  in  cycles  per  second. 
z  =  coefficient  of  hysteresis. 

z  varies  from  .001   to   .006  according  to  the  quality  of  the 
magnetic  material,  a  fair  value  for  transformer  sheets  being  .0022. 


98  ALTERNATING  CURRENT  MOTORS. 

The  alternation  of  the  flux  in  the  thin  sheets  (14  mils)  com- 
prising the  transformer  causes  the  generation  within  each  sheet 
of  a  minute  value  of  e.m.f.  which  tends  to  send  current  through 
the  conducting  material  of  the  sheet.  This  current  in  its  pas- 
sage through  the  sheets  follows  the  laws  common  to  all  electric 
circuits  and  produces  heat  proportional  to  the  square  of  its 
value  and  the  resistance  through  which  it  passes.  The  watts 
thus  dissipated  may  be  expressed  as 

_ 


10" 

where  d  =  thickness  of  sheets  in  centimeters, 
e  =  coefficient  of  eddy  loss. 

e  varies  with  the  specific  conductivity  of  the  core  material,  a 
fair  value  being  1.65. 

The  eddy  current  and  hysteresis  losses  being  similar  in  effect 
are  frequently  treated  as  one  quantity  under  the  term  core 
losses,  requiring  in  the  primary  coil  a  current  in  phase  with  the 
impressed  e.m.f.,  and  of  a  value  such  that  its  product  with  the 
e.m.f.  gives  the  core  loss  watts. 

While  the  value  of  the  flux  in  the  core  is  determined  almost 
exclusively  by  the  primary  e.m.f.  the  number  of  turns  and  the 
frequency,  quite  independent  of  the  permeability  of  the  magnetic 
path,  the  value  of  the  magnetomotive  force  to  produce  such  flux 
is  directly  dependent  upon  the  permeability  and  varies  inversely 
therewith.  A  convenient  method  for  obtaining  the  value  of  the 
magnetomotive  force  expressed  in  ampere  turns  is  found  from 

the  fact  that  one  ampere  turn  produces  -~  lines  per  centimeter 

cube  of  air.     From  this  fact  it  follows  that  one  ampere  turn  pro- 

1  o  5  1  1  A 
duces  --  j-  —  lines  in  a  material  of  permeability  //,  whose  length 

of  magnetic  path  is  /  centimeters  and  cross  sectional  area  A  sq. 
centimeters.  A  convenient  method  for  determining  the  exciting 
watts  from  the  volume  of  the  core  will  be  discussed  later. 

EQUIVALENT  ELECTRIC  CIRCUITS. 

For  the  magnetic  and  electric  circuits  of  a  transformer  as 
represented  in  Fig.  46  may  be  substituted  the  equivalent  electric 
circuits  shown  in  Fig.  47,  where  Rp  and  XP  are  the  primary  re- 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.      99 

sistance,  and  local  leakage  reactance,  while  Rs  and  Xs  are  the 
secondary  resistance  and  local  leakage  reactance,  the  shunted 
inductive  and  non-inductive  circuits  carrying  the  exciting  cur- 
rent and  core  loss  current  respectively.  These  are  the  true 
equivalent  circuits  of  a  transformer  based  upon  a  ratio  of  pri- 
mary to  secondary  turns  of  1  to  1.  If  the  primary  has  n  times 
as  many  turns  as  the  secondary,  then  the  same  equivalent 
circuits  may  be  used  to  represent  the  transformer,  if  the  actual 
secondary  resistance  and  local  leakage  reactance  be  multiplied 
by  n2  to  obtain  the  values  to  be  used  in  the  equivalent  circuits 
arid  the  real  secondary  load  current  be  divided  by  n. 

In  the  non-inductive  shunt  circuit  flows  the  current  to  supply 
the  core  losses.  If  these  losses  varied  as  the  square  of  the  in- 
ternal counter  e.m.f.,  that  is,  as  the  square  of  the  magnetic 
flux,  the  circuit  could  be  considered  as  composed  of  true  re- 


I>1 

FIG.  47. — Equivalent  Electric  Circuits  of  an  Ideal  Transformer. 

sistance.  Such,  however,  is  true  only  with  reference  to  the 
eddy  current  loss  and  is  not  directly  applicable  to  the  hysteresis 
loss,  but  the  assumption  of  the  existence  in  this  circuit  at  all 
times  of  a  current  whose  product  with  the  voltage  supplies  the 
core  losses  eliminates  any  error  due  to  treating  the  circuit  as 
being  of  pure  resistance. 

MODIFIED  ELECTRIC  CIRCUITS. 

It  is  obviously  possible  to  derive  readily  complete  equa- 
tions representing  the  performance  of  the  transformer  under 
various  conditions  of  load  by  the  use  of  the  circuits  shown  in 
Fig.  47,  when  proper  values  are  assigned  to  the  several  constants 
there  indicated.  There  may,  however,  be  introduced  in  the 
arrangement  of  the  circuits  a  slight  modification  which  in- 
volves no  measurable  error  and  yet  which  allows  the  performance 


100 


ALTERNATING  CURRENT  MOTORS 


of  the  transformer  to  be  represented  graphically  by  a  diagram 
whose  most  prominent  feature  is  its  simplicity.  In  Fig.  48, 
the  two  shunted  circuits  are  shown  as  connected  in  the  supply 
line  so  as  always  to  receive  the  full  value  of  the  impressed 
e.m.f.  The  magnitude  of  the  error  thus  produced  will  be  ap- 
preciated when  it  is  recalled  that  the  current  for  supplying  the 
core  losses  and  the  exciting  current  taken  by  a  transformer 
are  in  any  case  quite  small  and  the  assumption  that  the  com- 
bined value  of  such  small  currents  remain  constant  when  in 
reality  it  varies  inappreciably  (seldom  over  two  per  cent.) 
with  the  load  leads  to  a  truly  negligible  discrepancy  in  the 
results  thus  obtained. 

With  connections  made  as  indicated  in  Fig.  48,  the  current 
taken  by  each  of  the  three  circuits  will  flow  independently  of 


FIG.  48. — Practically  Exact  Representation  of  Circuits  of  a 
Transformer  or  Induction  Motor. 


the  currents  in  the  other  two  circuits  while  the  total  measurable 
primary  current  will  be  the  vector  sum  of  the  three  components. 
Methods  for  determining  the  value  of  the  current  for  supplying 
the  core  losses  and  the  exciting  current  have  been  given. 

CIRCLE  DIAGRAM  OF  CURRENTS. 

The  current  taken  by  the  load  flows  through  the  local  im- 
pedance of  both  the  primary  and  secondary  coils  and  is  unaffected 
by  the  presence  of  the  currents  in  the  other  shunted  circuits. 
If  the  external  load  circuit  be  strictly  non-inductive,  the  locus 
of  the  load  current  with  change  in  the  resistance  will  be  the  arc 
of  a  circle,  whose  diameter  is  the  ratio  of  the  primary  e.m.f. 
to  the  sum  of  the  local  reactances  of  the  primary  and  secondary 
coils.  (1  to  1  ratio.) 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    101 

Let  RL  be  any  chosen  value  of  load  resistance,  then  the  load 
current  will  be 


V  (RL 


and  its  phase  position  with  reference  to  the  primary  e.m.f.     EP 
will  be  such  that 

/)  ^  *-  p  ~T~  X  5 

sm  6  = 


V  (Rt 
hence 


which  when  Ep,  XP  and  X5  are  constants  is  the  polar  equation 
of  a  circle  having  diameter 


Knowing  the  values  of  the  three  component  currents  in  the 
branch  circuits  of  Fig.  48,  the  resultant  primary  current  may  be 
found  in  value  and  phase  position  as  their  vector  sum.  Since 
the  exciting  current  and  the  core  loss  current  do  not  change 
with  variation  in  the  load,  their  vector  sum  may  be  perma- 
nently recorded  as  the  no-load  primary  current  as  shown  at 
M  O  in  Fig.  49.  It  is  convenient  also  to  plot  at  once  the  vector 
of  the  load  current  O  P  in  position  to  give  at  once  the  resultant 
primary  current,  by  beginning  the  current  locus  O  P  C  at  the 
point  O  in  Fig.  49. 

A  study  of  the  construction  of  the  current  locus  of  Fig.  49 
will  show  that  at  any  chosen  load  current,  as  O  P,  M  P  is  the 
resultant  primary  current,  G  M  P  is  the  angle  of  lag  of  the 
primary  current  behind  the  e.m.f.  M  E,  and  the  ratio  of  M  G 
to  M  P  is  the  power  factor.  The  product  of  M  G  and  the  im- 
pressed primary  e.m.f.  is  the  input  to  the  transformer,  from 
which  if  the  core  losses  and  the  primary  and  secondary  copper 
losses  be  subtracted  the  output  may  be  obtained  and  the  effi- 
ciency determined.  The  output  divided  by  the  secondary 
current  gives  the  secondary  e.m.f.  from  which  the  regulation 
of  the  transformer  may  be  ascertained. 


102 


ALTERNATING  CURRENT  MOTORS, 


In  a  well  constructed  static  transformer,  the  primary  and 
secondary  coils  are  so  interspaced  that  magnetic  leakage  is 
•reduced  to  a  minimum,  so  that  XP  and  Xs  are  small  in  value 
and  the  diameter  of  the  current  locus  as  shown  in  Fig.  49  is 
correspondingly  enormous,  and  throughout  the  operating  range 
of  the  transformer  the  arc  of  the  circle  deviates  but  slightly 
from  a  straight  line  parallel  to  M  E.  Numerous  theoretical 
equations  are  available  for  determining  the  values  of  XP  and  Xs. 
Only  those  which  are  formed  upon  an  experimental  basis  in- 


FIG.  49. — Circular  Diagram  of  Currents. 

volving  the  use  of  the  exact  type  of  transformer  under  con- 
sideration are  found  to  give  results  in  conformity  to  observations 
under  test. 

On  account  of  the  fact  that  the  path  of  the  magnetic  lines  in 
the  core  of  a  static  transformer  pass  through  material  of  high 
permeability  a  relatively  small  value  of  exciting  magnetomotive 
force  is  required  and  due  also  to  the  low  value  of  the  core  losses 
the  no  load  current  of  such  a  transformer  is  correspondingly 
small  in  comparison  with  the  full  load  current.  A  type  of 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    103 

transformer  in  which  the  proportions  of  no  load  current  to  full 
load  current  is  quite  large  is  found  in  the  induction  motor.  In 
its  electrical  characteristics  an  induction  motor  is  essentially  a 
transformer  possessing  high  magnetic  leakage  due  to  the  separa- 
tion of  the  primary  and  secondary  windings  and  the  more  or 
Jess  complete  surrounding  of  each  group  of  coils  with  material 
of  high  permeability.  This  transformer  shows  also  a  high  value 
of  exciting  current  due  to  the  double  air  gap  in  the  magnetic 
circuit.  For  studying  the  characteristics  of  such  a  transformer 
the  circle  diagram  is  especially  valuable. 

Although  in  the  secondary  circuit  the  frequency  varies 
directly  with  the  slip,  and,  therefore,  for  constant  coefficient  of 
self-induction,  the  secondary  reactance  varies  in  direct  propor- 
tion with  the  slip,  and  in  general  the  secondary  resistance  is 
more  or  less  constant,  it  is  convenient  and  helpful  to  treat  the 
machine  as  a  stationary  transformer  in  which  the  secondary 
reactance  is  constant  and  the  resistance  varies  with  the  load. 
That  is  to  say,  the  performance  of  the  secondary  of  the  motor 
will  be  faithfully  represented  if  all  the  power  received  from  the 
primary  be  considered  as  dissipated  in  resistance  in  the  sec- 
ondary circuit,  the  slip  at  all  times  being  taken  as  unity,  and 
the  total  secondary  resistance  (conductance)  being  assumed  to 
be  varied  according  to  the  secondary  load,  or,  briefly,  the  in^ 
duction  motor  may  be  treated  in  all  respects  like  a  stationary 
transformer.  The  effect  upon  the  transformer  quantities  of 
increasing  the  speed  from  zero  to  synchronism  is  the  same  in 
all  respects  as  increasing  the  external  resistance  in  the  sec- 
ondary circuit  from  zero  to  infinite  value,  the  output  from  the 
motor  being  represented  in  any  case  as  the  power  lost  in  the 
fictitious  external  resistance.  These  facts  have  been  discussed  in 
a  preceding  chapter  and  they  need  not  be  further  discussed  here. 

The  diagram  of  Fig.  47,  which  shows  the  conventional  method 
of  representing  the  electric  circuits  of  a  transformer  which  has 
an  equal  number  of  turns  in  the  primary  and  secondary  wind- 
ings, is  based  on  the  assumptions  that  the  reluctance  of  the 
core  is  constant  for  all  densities  of  magnetisms  and  that  the 
iron  losses  vary  with  the  square  of  the  magnetic  density  in  the 
core.  It  is  obvious  that  neither  of  these  assumptions  is  abso- 
lutely correct  with  reference  to  any  commercial  stationary 
transformer,  but  it  is  true  that  the  errors  involved  in  any 


104         ALTERNATING  CURRENT  MOTORS. 

calculations  depending  upon  these  assumptions  are  practically 
negligible.  It  is  evident  that  in  an  induction  motor  the  re- 
luctance of  the  total  magnetic  path  is  much  more  nearly  con- 
stant than  that  in  a  transformer,  and  that  if  the  frictional 
and  windage  losses  be  included  with  the  iron  losses,  the  circuits 
shown  in  Fig.  47  will  serve  admirably  for  all  calculations  con- 
nected with  this  machine. 

In  the  diagram  of  Fig.  47,  RP  and  Xp  are  the  primary  re- 
sistance and  local  leakage  reactance,  while  Rs  and  Xs  are  the 
secondary  resistance  and  local  leakage  reactance,  the  shunted 
inductive  and  non-inductive  circuits  carrying  the  exciting  cur- 
rent and  the  core  loss  current,  respectively,  as  stated  previously. 
An  examination  of  Fig.  47  will  show  that  the  primary  current 
is  made  up  of  three  components;  the  quadrature  exciting  cur- 
rent, the  core  loss  current  and  the  load  current,  of  which  it  is 
the  vector  sum.  Under  operating  conditions  the  current  which 
flows  through  the  primary  coil  causes  a  drop  in  voltage  across 
the  local  primary  impedance  and  hence  the  internal  counter 
e.m.f.  decreases  with  increase  of  load,  and  there  is  a  decrease 
in  both  the  exciting  current  and  the  core  loss  current.  If  it  be 
assumed  initially  that  the  variation  in  the  values  of  these  cur- 
rents is  negligible  in  comparison  to  the  load  current  of  the 
machine,  the  treatment  becomes  much  simplified  and  yet  the 
true  conditions  are  fairly  well  represented. 

Fig.  48  shows  the  circuits  as  they  could  be  represented  on  the 
basis  of  the  latter  assumptions.  The  current  taken  by  each  of 
the  three  circuits  will  flow  independently  of  the  currents  in  the 
other  two  circuits,  while  the  total  measurable  primary  current 
will  be  the  vector  sum  of  the  three  components.  Only  one  of 
the  component  currents  varies  with  the  change  in  load,  and  its 
value  can  easily  be  determined  when  the  resistance  of  the  load 
circuit  is  known.  It  will  be  noted  that  the  load  circuit  con- 
ains  a  constant  reactance  (Xp  +  Xs)  in  series  with  a  variable 
resistance  (Rp  +  Rs  +  RL),  where  RL  is  the  fictitious  resistance 
of  the  load.  It  will  be  seen,  therefore,  that  the  current  which 
flows  through  this  circuit  under  a  constant  impressed  e.m.f., 
£,  can  be  represented  by  a  vector  whose  extremity  describes 
the  arc  of  a  circle  having  a  diameter  equal  to 

E 
Xp  +  Xs 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    105 

The  arc  O  P  C  of  Fig.  49  indicates  a  section  of  such  a  sec- 
ondary current  locus.  At  any  point,  P,  on  this  arc,  O  P  repre- 
sents the  secondary  current  both  in  value  and  phase  position. 
The  quadrature  exciting  current  is  represented  by  O  N,  while 
N  M  shows  the  core  loss  current  (to  supply  all  of  the  no-load 
losses).  The  vector  sum  of  these  three  currents,  M  P,  in  Fig. 
49,  is  the  primary  current,  while  the  angle  of  lag  of  the  current 
behind  the  circuit  e.m.f.  is  shown  by  N  M  P,  the  cosine  of 
which  is  the  power  factor.  The  power  component  of  the  pri- 
mary current  is  indicated  by  the  line  P  Q,  and  the  product  of 
this  with  the  circuit  e.m.f.  gives  the  input  to  the  motor. 

By  means  of  this  simple  circle  diagram,  the  construction  of 
which  is  based  on  somewhat  erroneous  assumptions,  the  com- 
plete performance  of  the  motor  may  be  determined  with  a 
degree  of  accuracy  which  seldom  need  be  exceeded  for  any  pur- 
pose of  designing  or  testing,  since  the  errors  introduced  are  of 
small  moment,  and  are  not  misleading;  moreover,  they  tend  to 
disappear  in  the  final  composite  results.  Thus  the  input  to 
the  secondary  at  any  chosen  current  may  be  found  as  the 
difference  between  the  primary  input  and  the  sum  of  the  pri- 
mary losses,  which  latter  include  the  easily  calculated  copper 
loss  and  the  approximated  "constant  "  losses.  The  secondary 
input  is  at  once  the  torque  in  "  synchronous  watts  ";  the  ratio 
of  the  secondary  copper  loss  to  the  secondary  input  is  the  slip, 
while  the  output  is  the  secondary  input  minus  the  secondary 
copper  loss.  As  will  be  shown  later,  the  various  quantities 
may  be  represented  graphically  by  simple  circular  arcs  and 
straight  lines. 

INTERNAL  VOLTAGE  DIAGRAM  OF  THE  INDUCTION  MOTOR. 

In  comparing  Fig.  48  on  which  the  simple  circle  diagram  of 
Fig.  49  is  based,  with  Fig.  47  on  which  a  true  diagram  should 
be  based,  it  will  be  observed  that  the  errors  involved  relate  merely 
to  the  quadrature  exciting  current  circuit  and  the  core  loss 
current  circuit;  the  voltage  across  these  circuits  is  not  con- 
stant, but  it  varies  with  the  load  current.  Since  that  portion  of 
the  drop  of  voltage  across  the  primary  impedance,  which  is 
due  solely  to  the  core  loss  current  and  the  exciting  current,  is 
quite  negligible  in  comparison  to  that  due  to  the  local  current, 
it  is  permissible  to  assume  that  the  voltage  at  B  D  in  Fig.  47 


106 


ALTERNATING  CURRENT  MOTORS. 


depends  entirely  upon  the  load  current.  This  assumption  is 
equivalent  to  neglecting  terms  of  higher  order.  These  may  be 
taken  into  consideration  graphically  without  difficulty,  but  the 
gain  by  so  doing  is  not  sufficient  to  justify  the  added  com- 
plications. 

The  internal  voltage  diagram  of  an  induction  motor  is  repre- 
sented in  Fig.  50,  where  A  D  is  the  impressed  primary  e.m.f., 
A  F  is  the  drop  through  the  primary  reactance,  B  F  is  the  drop 
through  the  primary  resistance,  and,  hence,  A  B  is  the  primary 
impedance  drop.  The  secondary  reactance  drop  is  B  H,  C  H 


FIGS.  50  and  51. — Modified  e.m.f.  and  Current  Loci  of  Poly- 
phase Induction  Motor. 

is  the  secondary  resistance  drop,  and  B  C  is  the  secondary  im- 
pedance drop.  C  D  is  the  voltage  consumed  in  the  fictitious 
external  load  resistance,  and,  hence,  it  the  secondary  e.m.f.; 
Es  in  Fig.  48  or  Fig.  47.  The  line  B  D  in  Fig.  50  is  the  e.m.f., 
EL  in  Fig.  48  or  the  voltage  across  B  D  in  Fig.  47. 

The  current  in  the  secondary  is  in  phase  with  G  D,  and  in 
quadrature  with  A  G.  The  angle,  A  G  D,  is  a  right  angle,  so 
that  the  point,  G,  describes  a  circle  with  its  center  on  the  line, 
A  D.  The  point,  F,  describes  a  circle,  API,  with  its  center 
at  point,  J,  on  line,  AID.  The  distance,  A  I,  bears  to  the 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    107 

distance,  A  D,  the  ratio  of  the  primary  leakage  reactance  to  the 
total  leakage  reactance  of  the  machine.  The  point,  B,  describes 
a  circle,  A  B  I L.  The  angle,  LAI,  has  a  tangent  equal  to 
the  ratio  of  the  primary  resistance  to  the  primary  leakage 
reactance. 

It  will  be  noted  from  Fig.  47  that  the  three  component  currents 
may  be  determined  at  once  when  the  secondary  load  current  is 
known  and  the  voltage  across  B  D  has  been  found.  Referring 
now  to  Fig.  52  and  remembering  that  the  core  loss  current  is  in 
phase  with  B  D  and  proportional  to  it,  it  will  be  seen  that  this 
current  can  be  represented  by  the  line,  D  T,  where  T  describes 
a  circle  having  its  center  on  the  line,  V  X,  the  angle,  D  V  X, 
being  equal  to  the  angle  I  A  L.  Similarly,  the  exciting  cur- 
rent, which  is  in  quadrature  with  B  D,  can  be  represented  by 
the  line,  D  S,  where  5  describes  a  circle  having  its  center  on  the 
line,  U  W,  the  angle,  D  U  W,  being  equal  to  the  angle,  I  A  L. 
D  U  and  D  V,  of  Fig.  52,  are  respectively  equal  to  D  U  and 
D  V  of  Fig.  50  or  to  0  N  and  N  M  of  Fig.  51. 

CORRECTED  CURRENT  Locus  OF  THE  INDUCTION  MOTOR. 

The  corrected  current  locus  of  the  induction  motor  is  shown 
in  Fig.  53,  where  the  arc,  O  P  R,  is  in  all  respects  the  same  as  the 
semicircle,  0  P  R,  in  Fig.  51.  The  line  O  P  in  Fig.  53  is  parallel 
to  the  line,  D  G,  in  Fig.  52,  but  it  varies  in  length  directly  with 
the  line,  A  G,  of  Fig.  52.  N  M  ot  Fig.  53  is  equal  to  D  T,  and 
is  parallel  and  proportional  to  B  D  of  Fig.  52.  0  N  is  equal  to 
D  S,  is  in  quadrature  to  and  proportional  to  B  D  of  Fig.  52. 
The  primary  current  is  represented  by  the  line,  M  P,  as  the 
vector  sum  of  0  P,  ON  and  N  M,  that  is,  as  the  resultant  of 
the  load  current,  the  quadrature  exciting  current  and  the  core 
loss  current. 

In  Fig.  53,  the  line  P  Q  is  the  power  component  of  the  pri- 
mary current,  the  product  of  which  with  the  impressed  e.m.f. 
gives  the  input  to  the  motor.  The  complete  performance  of 
the  machine  can  be  determined  in  a  manner  exactly  similar  to 
that  outlined  for  the  simple  circular  locus  of  Fig.  51  or  49. 

The  current  locus  in  Fig.  53  is  based  primarily  on  the  trans- 
former features  of  the  induction  motor,  and  it  is  inexact  only 
to  the  extent  to  which  the  motor  differs  from  a  transformer  in 
its  electrical  behavior.  The  frictional  loss  is  treated  as  a  core 


108 


ALTERNATING  CURRENT  MOTORS. 


loss,  and  hence  it  is  tacitly  assumed  that  this  loss  necessitates 
a  power  component  of  current  in  only  the  primary  winding. 
This  treatment  involves  no  error  with  reference  to  the  primary 
current,  but  it  neglects  a  certain  component  of  secondary  cur- 
rent, which,  however,  is  too  small  to  need  consideration. 

It  will  be  noted  from  Fig.  50  that  the  secondary  resistance 
has  no  effect  whatsoever  on  either  the  current  locus  shown  in 
Fig.  51,  or  that  shown  in  Fig.  53.  The  primary  resistance  has 
no  effect  on  the  secondary  current,  although  the  primary 


FIGS.  52  and  53. — Corrected  e.m.f .  and  Current  Loci  of  Poly- 
phase Induction  Motor. 

current  depends  somewhat  on  this  resistance.  If  the  primary 
resistance  were  negligible,  the  point  B  in  Fig.  52  would  follow 
the  circular  arc,  A  F  7,  and  both  the  angle,  D  U  W,  and  the 
angle,  DV  X,  would  reduce  to  zero;  the  general  form  of  the 
locus  of  Fig.  53  would  be  changed  only  slightly. 

From  the  facts  stated  above,  it  is  evident  that  the  primary 
current  cannot  be  represented  by  any  circle  howsoever  located, 
but  that  in  any  event  the  secondary  current  locus  is  a  true  circle 
For  many  purposes  where  extreme  accuracy  is  not  desired,  but 
where  some  information  is  wished  concerning  the  changes  in 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    109 

the  variables  connected  with  the  phenomena  of  an  induction 
motor  during  operation  an  approximate  graphical  diagram 
without  serious  errors  is  extremely  convenient.  It  is  believed 
that  the  simple  circular  locus  with  its  well  denned  but  practi- 
cally negligible  errors  possesses  peculiar  merit  in  this  respect. 

COMPLETE  PERFORMANCE  DIAGRAM  OF  THE  POLYPHASE  INDUC- 
TION MOTOR. 

A  simple  circular  locus  from  which  the  complete  performance 
of  a  polyphase  induction  motor  may  be  ascertained  at  once  is 
shown  in  Fig.  54.  At  any  point  P  on  this  locus,  the  line  M  P 
represents  the  primary  current,  while  the  angle,  E  M  P  is  the 
angle  of  lag  of  the  current  behind  the  primary  e.m.f.,  EM. 
The  line  0  P  shows  the  secondary  current,  both  in  value  and 


T  I'  I 

FIG.  54. — Simple  Circular  Current  Locus  of  a  Polyphase 
Induction  Motor. 

phase  position.  When  the  secondary  current  has  zero  value, 
that  is,  at  synchronous  speed,  the  primary  current  becomes 
equal  to  MO,  ON  being  its  "  wattless  "  component  and  N  M 
its  power  component  to  supply  all  of  the  no-load  losses.  When 
the  rotor  is  stationary,  the  secondary  current  assumes  some 
value  such  as  0  F,  and  the  primary  current  is  the  vector  sum 
of  0  F  and  OM  (not  drawn).  The  curve,  O  P  F,  is  the  arc 
of  a  circle  having  its  center  on  the  line  O  N  prolonged. 

Under  starting  conditions,  all  of  the  power  received  by  the 
motor  is  used  in  supplying  the  copper  and  core  losses  of  the 
machine.  The  line  F  I  is  the  power  component  of  the  primary 
current  at  starting,  and  hence,  by  the  use  of  the  proper  scale, 
it  may  represent  the  total  losses  of  the  machine  when  the  rotor 


110  ALTERNATING  CURRENT  MOTORS. 

is  stationary.  If  it  be  assumed  that  the  circuits  of  the  machine 
are  faithfully  represented  by  Fig.  48,  then  the  line  H  F  = 
(F  I  —  M  N)  of  Fig.  54,  shows  the  secondary  copper  loss  and 
the  increase  of  the  primary  copper  loss  over  its  (synchronous) 
no-load  value.  The  line  H  F  being  properly  divided  at  G,  G  H 
represents  the  increase  of  the  primary  copper  loss,  and  G  F 
the  secondary  copper  loss  for  the  current  O  F.  By  drawing 
from  the  point  O  a  straight  line,  0  G  J,  passing  through  the  point 
G,  the  complete  performance  of  the  machine  may  be  determined 
directly  from  inspection. 

If  from  any  point  P  on  the  circular  locus  a  line  be  drawn  per- 
pendicular to  the  diameter,  O  K,  the  following  quantities  may 
be  observed  at  once: 

M  P  is  the  primary  current, 
E  M  P  is  the  primary  angle  of  lag, 
cos  E  M  P  is  the  power  factor, 

O  P  is  the  secondary  current, 
P  T  is  the  total  primary  input, 
T  S  is  the  "  constant  "  losses  of  the  machine, 
R  S  is  the  added  primary  copper  loss, 
R  T  is  the  total  primary  losses, 
P  R  is  the  total  secondary  input,  in  watts, 
P  R  is  the  torque  in  synchronous  watts, 
Q  R  is  the  secondary  copper  loss, 
QR  +  PR  is  the  slip,  with  synchronism  as  unity, 
QP  +  P  Ris  the  speed, 

Q  P  is  the  output, 
Q  P  +  P  E  is  the  efficiency. 

The  maximum  power  factor  occurs  when  the  point  P  is  at  A, 
where  the  line  M  P  becomes  tangent  to  the  circle. 

The  maximum  output  occurs  when  P  is  at  B,  the  point  of 
tangency  of  a  line  drawn  parallel  to  0  F. 

The  maximum  torque  occurs  when  P  is  at  C,  the  point  of 
tangency  of  a  line  drawn  parallel  to  0  G.  The  current  which 
is  required  to  give  maximum  torque  varies  somewhat  with  the 
primary  resistance,  but  it  is  independent  of  the  secondary  re- 
sistance, although  the  speed  at  which  the  maximum  torque  is 
obtained  depends  largely  on  the  value  of  the  secondary  resist- 
ance. The  secondary  resistance  required  to  give  maximum 
torque  at  starting  bears  to  the  assumed  constant  primary  re- 
sistance the  ratio  of  G'  C  to  G'  H'  of  Fig.  54. 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    Ill 

The  proof  of  the  accuracy  of  the  diagram  in  representing  the 
value  and  phase  positions  of  the  primary  and  secondary  currents 
for  the  circuits  as  shown  in  Fig.  48  was  given  above,  and  it 
need  not  here  be  repeated.  That  the  various  other  quantities 
are  accurately  represented  as  indicated  may  be  shown  as  follows: 

Denoting  as  a  the  angle  P  0  K  of  Fig.  54,  it  will  be  seen  that 
O  S  =  O  P  cos  a,  and  that  OP  =0  K  cos  a.  Hence,  05  = 
O  K  cos2  a,  or  0  S  =  O  P2  +  O  K.  The  interpretation  of  the  last 
equation  is  that,  as  the  point  P  moves  around  the  circle,  the 
line  0  S  is  at  all  times  proportional  to  the  square  of  the  line  0  P. 
That  is  to  say,  the  line  O  5  is  proportional  to  the  secondary 
copper  loss  or  to  the  increase  in  the  primary  copper  loss  over 
its  no-load  value.  Under  starting  conditions,  the  secondary  and' 
the  added  primary  copper  losses  are  represented  by  F  G  and  G  H ; 
and  at  any  point  P,  the  corresponding  losses  must  bear  to.  F  G 
and  to  G  H  the  ratio  of  0  S  to  O  H.  Therefore,  the  secondary 
and  the  added  primary  copper  losses  are  accurately  shown  by 
the  lines  Q  R  and  R  S,  respectively. 

That  the  ratio  of  the  secondary  copper  loss  to  the  total  secondary 
input  is  equal  to  the  slip  has  already  frequently  been  metioned. 
It  seems  desirable,  however,  in  this  connection  to  call  attention 
to  the  fact  that  this  ratio  is  a  true  measure  of  the  slip  whether 
the  magnetism  of  the  machine  remains  constant  or  not,  and 
that  the  accuracy  of  the  determination  of  the  slip  by  this  method 
is  in  no  wise  affected  by  the  substitution  of  the  modified  circuits 
of  Fig.  48  for  the  exact  circuits  of  Fig.  47.  Thus,  if  the  secondary 
copper  loss  is  determined  without  error,  and  the  secondary  input 
is  known,  both  the  speed  and  the  torque  may  be  ascertained  with 
precision.  It  is  seen,  therefore,  that  any  errors  introduced 
must  relate  to  either  the  currents  or  the  losses. 

If  the  points  O  and  F  of  Fig.  54  are  obtained  from  an  actual 
test  on  a  machine,  it  is  evident  that  the  circle  diagram  as  con- 
structed must  be  at  least  approximately  correct  for  the  primary 
current  locus.  Under  starting  conditions  the  power  received 
by  the  machine  is  accurately  represented  by  the  line  F  I.  If 
the  distance  F  G  be  drawn  equal  to  the  easily  determined 
"  added  "  primary  copper  loss,  the  distance  G  H  must  represent 
the  secondary  copper  loss  with  a  fair  degree  of  accuracy. 

It  is  especially  worthy  of  note  that  under  starting  conditions 
and  at  synchronous  speed  the  errors  are  eliminated,  and  through- 


112  ALTERNATING  CURRENT  MOTORS 

out  the  operating  range  of  the  motor  the  various  errors  tend  to 
cancel  each  other.  Even  in  extreme  cases,  where  the  (syn- 
chronous) no-load  triangle,  O  M  N,  is  large  in  comparison  with 
the  circle  diagram,  the  errors  are  relatively  small  and  for  most 
practical  purposes  may  well  be  neglected. 

It  is  not  possible  to  obtain  absolute  accuracy  in  a  simple 
diagram  of  an  induction  motor.  Moreover,  it  is  unnecessary  to 
construct  a  diagram  with  a  degree  of  accuracy  greater  than  that 
which  can  be  employed  with  it  in  scaling  off  the  various  values. 
The  principal  advantage  to  be  found  in  the  graphical  method  of 
treating  induction  motor  phenomena  resides  in  the  fact  that  by 
the  use  of  a  simple  diagram  one  is  able  to  follow  optically,  and 
thus  mentally,  the  changes  which  take  place  throughout  the 
operation  of  the  machine,  while  in  the  manipulation  of  algebraic 
formulas,  which  can  be  used  for  absolute  accuracy,  one  is  apt 
to  find  himself  more  or  less  in  the  dark  concerning  these  changes. 

The  above  description  refers  to  the  locus  of  the  primary  and 
secondary  currents  of  a  polyphase  induction  motor.  It  is  de- 
sirable to  describe  also  the  method  by  which  a  similar  diagram 
may  be  used  with  a  single-phase  induction  motor. 

COMPARISON  OF  SINGLE-PHASE  AND  POLYPHASE  MOTORS. 

The  chief  difference  between  a  single-phase  and  a  polyphase 
induction  motor  resides  in  the  character  of  the  magnetic  fields 
of  the  two  machines.  At  synchronous  speed  each  machine 
possesses  a  true  revolving  field.  At  standstill,  however,  while 
the  magnetic  field  of  the  polyphase  motor  revolves  synchron- 
ously and  is  of  more  or  less  constant  strength,  the  field  of  the 
single-phase  machine  is  unidirectional  in  space  and  alternating 
in  value. 

If  when  the  rotor  of  a  polyphase  motor  is  stationary  a  circuit 
be  opened  so  that  current  flows  through  only  two  leads  of  the 
machine,  it  will  be  found  that  the  total  volt-amperes  taken  by 
the  machine  decrease  to  about  one-half  of  the  former  value, 
the  power  factor  being  practically  unchanged.  That  the  mag- 
netomotive force  of  the  current  in  each  phase  winding  of  a  two- 
phase  motor  when  the  rotor  is  stationary  produces  a  flux  which 
(for  constant  reluctance  of  the  core)  acts  as  though  the  flux 
due  to  the  other  current  in  the  winding  were  not  present  may 
be  seen  at  once  without  proof.  It  will  be  appreciated,  also, 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    113 

that  the  currents  produced  in  the  secondary  by  two  alternating 
fluxes  which  are  in  electrical  space  quadrature  do  not  interfere 
one  with  the  other,  so  that  the  current  in  each  primary  winding 
flows  just  as  though  the  other  primary  current  did  not  exist. 
Thus  the  "  equivalent  single-phase  "  starting  current  of  a  two- 
phase  motor  is  just  twice  that  of  the  same  motor  when  only 
one  phase  winding  is  used;  the  power  factor  is  the  same  in  the 
two  cases.  Both  experimental  and  theoretical  investigations 
show  that  the  "  equivalent  single-phase  "  starting  current  of  a 
three-phase  motor  is  also  equal  to  twice  the  current  which  flows 
through  two  leads  when  the  third  lead  is  interrupted. 

If,  when  the  rotor  of  a  polyphase  induction  motor  is  revolving 
synchronously  a  primary  circuit  of  the  machine  be  opened,  it 
will  be  found  that  the  current  flowing  through  the  remaining 
leads  increases  somewhat  but  that  the  total  volt-amperes  taken 
by  the  machine  remain  practically  constant,  and  the  power- 
factor  is  practically  unaltered  (the  power  component  of  the 
equivalent  single-phase  current  increases  to  a  small  extent  while 
the  wattless  component  decreases  slightly).  The  action  of  the 
machine  at  synchronous  speed  is  attributable  to  the  continued 
existence  of  a  revolving  magnetic  field  or  practically  constant 
strength  which  requires  a  definite  component  of  current  in 
phase  with  the  voltage  to  supply  the  losses  and  another  com- 
ponent in  quadrature  with  the  voltage  to  supply  the  "  quadrature 
watts  "  for  excitation.  A  subsequent  chapter  will  explain  the 
distribution  of  current  in  the  secondary  conductor,  and  will 
show  in  what  manner  the  "  quadrature  watts"  for  the  "  speed- 
field"  are  supplied  by  the  primary  exciting  magnetomotive 
force. 

When  the  rotor  of  a  polyphase  motor  is  revolving  synchron- 
ously, the  secondary  current  has  a  negligible  value.  In  the 
single-phase  motor,  however,  the  secondary  current  at  synchron- 
ous speed  has  a  value  such  that  its  magnetomotive  force  produces 
in  electrical  space  quadrature  with  the  main  alternating  field 
through  the  primary  coil,  a  field  which  is  equal  in  value  and 
in  time  quadrature  with  the  main  field.  The  value  of  the  main 
field  is  determined  by  the  primary  e.m.f.  just  as  is  true  in  any 
transformer,  while  the  field  which  is  in  quadrature  both  in  time 
and  in  space  therewith  depends  for  its  value  both  upon  the 
"  transformer  field  "  and  upon  the  speed  of  the  rotor;  the  two 


114  ALTERNATING  CURRENT  MOTORS. 

fields  are  equal  in  effective  value  at  synchronous  speed  and  at 
other  speeds,  the  "  speed  field  "  is  equal  to  the  "  transformer  " 
field  multiplied  by  the  speed.  Thus  the  "  speed- field  "  com- 
ponent of  the  secondary  current  varies  with  the  speed  and  is 
zero  at  standstill. 

ELECTRIC  CIRCUITS  OF  THE  SINGLE-PHASE  INDUCTION  MOTOR. 

The  circuits  of  a  single-phase  induction  motor  can  be  repre- 
sented with  a  fair  degree  of  accuracy  if  the  primary  and  sec- 
ondary resistances  and  the  leakage  reactances  be  arranged  as 
shown  in  Fig.  55a,  the  "  transformer  field  "  and  "  speed  field  " 
exciting  circuits  being  connected  as  indicated.  The  current 
taken  by  the  load  and  that  used  to  produce  the  "  speed  field  " 
pass  through  both  the  primary  and  the  secondary  coils,  while 
the  current  required  for  the  "  transformer  field  "  flows  through 


FIG.  55 A. — Practically  Exact  Representation  of  Circuits  of 
a  vSingle-phase  Induction  Motor. 

only  the  primary  coil.  When  the  load  circuit  is  opened,  that 
is,  at  synchronous  speed,  the  "  speed  field  "  current  and  the 
"  transformer  field  "  current  are  practically  equal  in  value. 
When  the  resistance  of  the  load  circuit  is  zero,  that  is  at  stand- 
still, the  "  speed  field  "  current  is  zero,  and  the  current  which 
flows  through  the  coils  of  the  machine  acts  as  though  the  "  speed 
field  "  circuit  were  not  present. 

It  is  to  be  noted  especially  that  the  decrease  in  the  "  speed 
field  "  current  below  the  value  of  the  "  transformer  field  "  cur- 
rent is  attributable  to  the  variation  of  the  rotor  speed  from 
synchronism  and  not  to  the  drop  in  voltage  across  the  secondary 
winding,  which  is  caused  by  the  load  current.  The  "  secondary 
load  "  current  fiows  in  electrical  space  quadrature  to  the  "  speed 
field  "  current,  and  the  two  currents  in  no  way  interfere  with 
each  other.  The  statements  just  made  relate  exclusively  to 
the  current  whose  magnetomotive  force  produces  the  "  speed 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.     115 

field,"  which  current,  on  account  of  its  electrical  space  position, 
does  not  react  in  any  way  upon  the  "  transformer  field."  The 
secondary  carries  also  another  component  of  current  in  addition 
to  the  load  current.  The  time-phase  position  and  the  electrical 
space  positions  of  this  component  are  such  that  its  magneto- 
motive force  tends  directly  to  decrease  the  "  transformer 
field  ";  thus,  it  acts  like  a  "  wattless  "  secondary  current.  It 
is  this  latter  component  of  secondary  current  which  is  represented 
in  the  circuit  diagram  of  Fig.  55a.  This  component  bears  to  the 
actual  "  speed  field  "  current  (approximately)  the  ratio  of  the 
actual  rotor  speed  to  the  synchronous  speed.  Thus  the  voltage 
impressed  upon  the  "  speed  field  "  circuit  of  Fig.  55ais  (approx- 
imately) equal  to  that  impressed  upon  the  "  transformer  field  " 
circuit  multiplied  by  the  square  of  the  speed,  synchronism  being 


FIG.  SOB. — Modified  Representation  of  Circuits  of  a  Single- 
phase  Induction  Motor. 


taken  as  unity.     These  facts  will  be  discussed  more  fully  in  a 
subsequent  chapter. 

Although  it  is  possible  to  construct  primary  and  secondary 
current  loci  based  on  the  circuits  shown  in  Fig.  55a,  the  problem 
of  dealing  with  the  value  and  phase  positions  of  the  currents  is 
greatly  simplified  without  involving  a  detrimental  loss  of  accu- 
racy by  using  the  modified  arrangement  of  circuits  indicated 
in  Fig.  55b. 

COMPLETE  PERFORMANCE  DIAGRAM  OF  THE  SINGLE-PHASE  IN- 
DUCTION MOTOR. 

The  current  diagram  for  the  circuits  shown  in  Fig.  55b  is 
given  in  Fig.  56,  where  M  N  is  the  power  and  O  N  the  wattless 
component  of  the  primary  current  at  synchronous  no  load,  while 
F  I  is  the  power  and  I  M  the  wattless  component  of  the  pri- 
mary current  at  standstill.  The  curve  0  P  F  K  is  an  arc  of  a 


116 


ALTERNATING  CURRENT  MOTORS. 


circle  having  its  center  on  the  line  O  N  prolonged.  O  L  is  the 
"  speed  field  "  current  (assumed  constant  in  the  diagram,  but 
properly  accounted  for  in  the  computations) .  L  M  is  the 
"transformer  field"  current,  while  O  M  is  the  total  primary 
current  at  synchronism. 

The  line  F  I  drawn  perpendicular  to  M  T  I  represents  the 
total  loss  of  the  machine  at  standstill — the  proper  scale  being 
used.  HI  indicates  the  so-called  "constant"  losses,  while 
F  H  shows  the  sum  of  the  "  added  "  primary  and  secondary 
copper  losses.  If  the  distance  G  H  be  laid  off  to  represent 
accurately  the  easily  determinable  "  added  "  primary  copper 
loss,  then  F  G  shows  the  "added"  secondary  copper  loss. 
Straight  lines  being  drawn  to  join  the  point  F  and  the  point  G 
with  O  of  Fig.  56,  if  from  any  point  P  on  the  circular  arc  0  P  F  K 


FIG.  56. — Current  Locus  of  Single-phase  Induction  Motor. 

a  perpendicular  be  dropped  to  the  line  M  I  the  following  values 
may  be  taken  at  once  from  the  diagram: 

O  P  is  the  "  added  "  component  of  the  primary  current, 
0  Pis  also  the  "  added  "  component  of  the  secondary 
current, -- 

P  L  is  the  total  secondary  current, 
P  M  is  the  total  primary  current, 
Cos  E  M  P  is  the  power  factor, 
P  T  -f-  P  M  is  the  power  factor, 

S  T  is  the  "  constant  "  losses  of  the  machine, 
R  S  is  the  "  added  "  primary  copper  loss, 
R  T  is  the  total  primary  losses  (including  "  speed  "  field 
excitation  current  loss  in  secondary), 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    117 

QRis  the  "  added  "  secondary  copper  loss, 
Q  T  is  the  total  losses  of  the  machine, 
P  T  is  the  input  to  the  machine, 
Q  P  is  the  output, 
Q  P  +  P  T  is  the  efficiency, 

P  R  is  the  total  input  to  the  secondary  (excluding  the 
"  speed  field  "  excitation  current  loss). 
(PQ-r-P  R)*  is  the  speed  with  synchronism  as  unity, 
(PQXP^)Hs  the  torque  in  synchronous  watts. 

The  representation  of  each  of  the  quantities  listed  above, 
with  the  exception  of  the  speed  and  the  torque,  will  be  appre- 
ciated at  once  from  a  comparison  of  Fig.  55b  with  Fig.  56, 
combined  with  a  review  of  the  preceding  chapter. 

SPEED  AND  TORQUE  OF  THE  SINGLE-PHASE  INDUCTION  MOTOR. 

The  speed  and  the  torque  can  be  ascertained  in  an  extremely 
simple  manner  as  follows:  The  "  speed  "  field  is  under  any 
chosen  condition  equal  to  the  "  transformer  "  field  multiplied 
by  the  speed.  Now  the  torque  is  proportional  to  the  product 
of  the  "  speed  "  field  and  that  component  of  the  secondary 
current  which  is  in  time  phase  with  it  and  which  crosses  the 
core  at  the  same  mechanical  position  along  the  air  gap  as  that 
occupied  by  the  "  speed  "  field.  This  component  of  the  sec- 
ondary current  is  in  time  quadrature  with  the  "  transformer  " 
field,  and  it  has  a  value  such  that  its  product  with  the  primary 
e.m.f.  (for  a  unity  ratio  machine)  represents  the  total  power 
received  by  the  secondary — exclusive  of  the  loss  due  to  the 
secondary  excitation  current.  A  little  study  will  show,  there- 
fore, that  if  the  "  speed  field  "  were  equal  to  the  "  transformer 
field,"  the  torque  in  "  synchronous  watts  "  would  be  equal  to 
the  secondary  input  (excluding  the  excitation  loss).  Since  the 
"  speed  field  "  varies  directly  with  the  speed,  it  is  seen  at  once 
that  the  torque,  D,  is  equal  to  the  secondary  input,  W8,  multi- 
plied by  the  speed,  5.  Thus, 

D  =  S  Ws.  (1) 

The  torque  is  also  equal  to  the  output  W0,  divided  by  the 
speed,  therefore 

D  =  W«  +  S  (2) 

hence 

S  =  (W0  -*-  Ws)*  (3) 


118  ALTERNATING  CURRENT  MOTORS. 

That  is  to  say,  in  a  single-phase  induction  motor  the  speed  is 
equal  to  the  square  root  of  the  secondary  efficiency.  When  the 
speed  varies  only  a  few  per  cent,  from  synchronism  the  slip  is 
equal  to  one-half  of  the  secondary  loss  expressed  in  per  cent., 
as  was  pointed  out  by  Mr.  B.  A.  Behrend  on  page  884  of  the 
issue  of  the  Electrical  World  and  Engineer  for  Dec.  8,  1900. 
Thus  at  a  speed  of  .98  the  secondary  efficiency  is  .9604;  the 
slip  is  .02;  the  loss  is  .0396.  It  is  interesting  to  observe  in  this 
connection  that  in  a  polyphase  induction  motor  the  speed  is 
equal  directly  to  the  secondary  efficiency. 

Combining  equations  (1)  and  (3)  above,  it  is  found  that  the 
torque  has  the  following  value: 

D  =  (W0XWS)*  (4) 

That  is  to  say,  the  torque  is  equal  to  (PQxPR)^  from 
Fig.  56,  as  used  above. 

It  is  especially  worthy  of  note  that  the  speed  and  torque 
as  here  determined  are  not  affected  by  the  substitution  of  the. 
modified  circuits  of  Fig.  55b  for  the  more  nearly  exact  circuits 
of  Fig.  55a.  The  method  here  outlined  gives  correct  results 
both  at  synchronism  and  at  standstill,  and  at  other  intermediate 
speeds  the  slight  errors  introduced  are  of  both  positive  and 
negative  values,  and  they  tend  to  cancel  in  the  final  results. 

On  account  of  the  fact  that  at  speeds  below  synchronism 
there  is  a  slight  decrease  in  the  "  transformer- field  "  current 
and  a  large  decrease  in  the  "  speed-field  "  current  (as  it  reacts 
upon  the  primary),  while  Fig.  56  assumes  both  of  these  currents 
to  be  constant,  the  operating  power  factor  of  a  single-phase 
motor  is  somewhat  greater  than  that  shown  in  Fig.  57.  The 
discrepancy  is  appreciable  only  in  those  cases  where  the  "  syn- 
chronous no  load  "  current  is  large  in  comparison  with  the 
starting  current.  Thus  Fig.  57  gives  the  power  factor  accu- 
rately for  large  motors,  but  small  motors  will  show  better  power 
factors  than  there  indicated. 

It  is  instructive  to  compare  the  performance  of  a  certain 
polyphase  motor,  when  operated  normally,  with  that  of  the 
same  machine  when  used  as  a  single-phase  motor.  Using 
"  equivalent  single-phase  "  quantities  throughout,  the  polyphase 
starting  current  of  the  motor,  whose  single-phase  circle  dia- 
gram is  shown  by  the  arc  O  P  F  K  in  Fig.  56,  would  be  repre- 


TRANSFORMER  FEATURES  OF  INDUCTION  MOTOR.    119 

sented  by  the  line  M  F  Ff  (not  completely  drawn)  having  a 
length  equal  to  twice  that  of  the  line  M  F  (not  drawn).  The 
polyphase  current  locus  is  a  circle  passing  through  F'  and  0, 
its  center  being  on  the  line  O  N  prolonged.  If  the  machine  is 
operated  as  a  single-phase  motor  at  a  certain  primary  current, 
such  as  shown  by  M  P  in  Fig.  56,  the  output  is  P  Q,  as  noted 
above.  If  the  same  output  is  to  be  obtained  when  the  machine 
is  operated  polyphase,  then  the  equivalent  single-phase  value 
of  the  polyphase  current  will  be  M  P'  (not  drawn),  the  line  P  P' 
being  (practically)  parallel  with  the  line  O  F.  Thus  the  volt- 
amperes  input  as  a  single-phase  machine  is  greater  than  as  a 
polyphase  machine  in  the  ratio  of  M  P  to  M  P '  and  the  power 
factor  is  less  in  the  ratio  of  Cos  E  M  P  to  Cos  E  M  P';  the 
losses  are  also  greater. 

CAPACITIES  OF  SINGLE-PHASE  AND  POLYPHASE  MOTORS. 

Although  the  circular  diagram  as  developed  above  is  ap- 
plicable to  all  types  of  single-phase  induction  motors,  the  com- 
parison just  made  refers  exclusively  to  polyphase  motors  and 
the  same  motors  used  on  single-phase  circuits.  The  comparison 
between  single-phase  and  polyphase  machines  is  not  quite  so 
unfavorable  to  the  former  when  each  machine  is  designed  pri- 
marily for  its  particular  work.  When  a  polyphase  motor  is 
operated  as  a  single-phase  machine,  only  a  portion  of  the  pri- 
mary copper  is  fully  employed;  evidently  a  greater  output  can 
be  obtained  by  altering  the  inter-connections  of  the  coils  so 
as  to  use  all  of  the  copper. 

With  an  induction  motor  having  uniformly  distributed  coils, 
when  the  iron  is  subjected  to  the  same  magnetic  density  and 
frequency,  and  the  same  current  density  is  used  in  the  copper, 
the  output  varies  largely  with  the  groupings  of  the  coils.  Thus 
it  may  be  shown  that  with  such  a  motor  the  volt-ampere  input 
can  be  represented,  relatively,  by  the  periphery  of  a  polygon 
having  sides  equal  in  number  to  the  number  of  groups  per  pair 
of  poles,  of  which  polygon  the  circumscribing  circle  represents 
the  volt-ampere  input  for  infinite  groups,  and  double  the  diam- 
eter represents  the  input  to  the  single-phase  motor.  Giving  to 
the  diameter  of  the  circumscribing  circle  an  arbitrary  value  of 
unity,  the  inputs  to  the  machine  for  various  groupings  of  coils 
are  as  follows: 


120  ALTERNATING  CURRENT  MOTORS. 

Number  of  Groups.     Type  of  Machine.  Volt-Ampere  Input 

2  Single-phase  2 . 000 

3  Three-phase  2.598 

4  Four- phase  2.828 
(Two-phase) 

6  Six-phase  3.000 

(Three-phase) 

Since  the  coils  of  commercial  two-phase  induction  motors 
are  grouped  similarly  to  those  of  a  four-phase  machine  and 
the  coils  of  a  three-phase  motor  are  arranged  similarly  to  those 
of  a  six-phase  machine,  a  three-phase  motor  has  a  volt-ampere 
input  1.061  times  that  of  a  two-phase  motor  (on  the  basis  of 
equality  losses),  while  the  volt-ampere  input  of  a  single-phase 
motor  is  .707  times  that  of  an  equivalent  two-phase  machine. 
The  facts  upon  which  these  statements  are  based  are  discussed 
more  fully  in  the  next  chapter. 


CHAPTER  X.    , 
MAGNETIC  FIELD  IN  INDUCTION  MOTORS. 

POLYPHASE  MOTORS. 

In  construction  an  induction  motor  possesses  as  primary 
windings,  coils  placed  mechanically  around  a  core  and  sepa- 
rated as  to  polarization  effects  by  the  same  number  of  angular 
degrees  as  the  currents,  which  flow  in  the  individual  coils, 
differ  in  electrical  time  degrees,  a  two-pole  model  being  assumed. 
Thus  in  a  two-phase  (or  quarter-phase)  machine  the  coils  would 
be  located  90  degrees  one  from  the  other,  and  in  a  three-phase 
motor  the  angular  spacing  would  be  60  degrees  (120  degrees). 

Consider  a  two-polar  quarter-phase  machine  upon  the  sepa- 
rate windings  of  which  there  are  impressed  e.m.fs.  in  time 
quadrature.  The  e.m.f.  at  each  coil  demands  that  at  each 
instant  the  resultant  magnetism  threading  that  coil  have  a 
certain  definite  value  such  that  its  rate  of  change  generates  in 
the  coil  the  proper  value  of  counter  e.m.f.,  just  as  is  true  in  any 
stationary  transformer.  No  action  which  takes  place  within 
the  machine  can  rob  the  primary  coils  of  this  transformer 
feature.  Assume  that  the  flux  (and  e.m.f.)  follows  a  sine  law 
of  change  of  value  with  reference  to  time  and  let  (j>  be  the  max- 
imum value  of  flux  demanded  by  the  e.m.f.  of  each  coil.  Then 
at  a  given  instant  the  e.m.f.  in  coils  1  and  2  expressed  in  c.g.s. 
units  will  be,  for  N  effective  turns. 

d  d 

e^  =  N  -      sin  o>  /  and 


^2  =  N  .g    sin    a*  t  -     -  hence 

el  =  N  oj  (f>  cos  w  t 

e2  =  N  to  <f)  sin  cu  t         or  at 

any  certain  time  t,  the  flux  demanded  by  the  e.m.fs.  must  have 

121 


122  ALTERNATING  CURRENT  MOTORS. 

a  value  of  $  cos  o>  t  threading  coil   1 ,  and  <j>  sin  o>  t  threading 
coil  2. 

In  commercial  induction  motors  the  coils  of  one  phase  winding 
overlap  those  of  the  other,  each  winding  being  distributed  over 
an  area  inversely  proportional  to  the  number  of  primary  phases. 
The  distributed  character  of  the  windings  is  such  that  the  flux 
which  threads  one  winding  simultaneously  threads  the  other, 
the  distribution  of  the  flux  alone  determining  the  actual  effective 
value  threading  each  coil  at  each  instant.  This  feature  of  the 
winding  combined  with  a  slight  value  of  modifying  current  in 
the  closed  conductors  of  the  secondary  winding  is  such  that 
at  any  time  t  the  fluxes  in  the  two  phase  motor  core  have  values 
$  cos  oj  t  and  <£  sin  cu  t  so  distributed  as  to  give  a  resultant  of 


cos2  co  t+(j>2  sin2  wt  =  <j). 

That  is  to  say,  the  resultant  core  flux  is  constant  in  value, 
but  varies  in  position,  producing  the  so-called  revolving  field. 
This  field  travels  at  a  speed  termed  "  synchronous,"  as  found 
"by  the  ratio  of  alternations  per  unit  time  to  the  number  of 
poles  of  the  machine. 

Within  this  revolving  field  is  placed  the  secondary  winding 
which  in  most  cases  is  closed  upon  itself  either  directly  or  through 
certain  variable  resistance.  When  the  rotor  is  traveling  at  syn- 
chronous speed,  the  secondary  conductors  are  not  subjected  to 
change  in  flux  except  to  the  slight  extent  due  to  the  irregulari- 
ties of  the  magnetic  field,  so  that  at  this  speed  only  the  modify- 
ing current  previously  referred  to  flows  in  the  secondary,  and 
this  current  tends  to  cause  the  revolving  field  to  have  a  truly 
constant  value. 

In  studying  the  internal  actions  of  the  induction  motor,  it 
is  helpful  to  consider  that  there  exists  a  revolving  field  of  a 
certain  value  traveling  at  synchronous  speed;  that  this  field 
cuts  the  primary  conductors  at  synchronous  speed  and  at  a  rate 
to  generate  therein  the  proper  counter  e.m.f.  and  that  the 
secondary  conductors  cut  across  this  field  at  a  rate  depending 
upon  the  difference  in  the  speeds  of  the  rotor  and  of  the  revolv- 
ing field,  that  is  upon  the  "  slip  "  from  synchronism.  Ac- 
cording to  this  conception  the  effective  value  of  the  e.m.f. 
counter  generated  in  each  conductor  on  the  face  of  the  primary 
core  will  be  the  same  irrespective  of  its  mechanical  position, 


MAGNETIC  FIELD.  123 

but  the  time-phase  position  of  this  e.m.f.  will  vary  directly  with 
the  location  on  the  core.  When  a  number  of  conductors  are 
connected  in  series  to  form  a  primary  coil,  the  effective  value 
of  the  resultant  e.m.f.  counter  generated  therein  will  be  the 
vector  sum  of  the  e.m.f.  of  the  individual  conductors.  From 
this  fact  may  be  determined  the  core  flux  necessary  to  give  a 
certain  counter  e.m.f.  with  a  certain  distribution  of  the  primary 
conductors. 

It  is  the  purpose  of  the  present  chapter  to  discuss  the  facts 
upon  which  these  statements  are  based,  to  show  that  the  space 
distribution  of  the  revolving  magnetic  flux  follows  a  sine  law, 
and  to  outline  the  method  by  which  the  implied  considerations 
can  be  utilized  in  the  treatment  of  induction  motor  phenomena. 

In  explanation  of  several  of  the  terms  used  below  it  should  be 
stated  that,  in  dealing  with  the  magnetic  field  in  induction 
motors  of  the  multipolar  type,  it  is  necessary  to  distinguish  be- 
tween "  electrical -time  "  degrees,  "  electrical-space  "  degrees 
and  "  mechanical-space  "  degrees.  Thus,  in  a  four-pole  machine, 
90  electrical-space  degrees  correspond  to  45  mechanical-space 
degrees.  Two  fluxes  may  be  said  to  be  in  electrical-time  quad- 
rature when  one  reaches  its  maximum  90  time  degree  (J  cycle) 
before  or  after  the  other.  Independent  entirely  of  their  time- 
phase  position,  they  would  be  in  electrical  space  quadrature  if 
their  mechanical  positions  in  the  air-gap  of  a  four-pole  machine 
were  displaced  45  mechanical-space  degrees  one  from  the  other. 

Numerous  writers  when  discussing  the  flux  in  the  air-gap  cf 
induction  motors  have  stated  in  substance  or  have  implied  by 
their  method  of  treatment  that  the  space  distribution  of  the 
magnetism  depends  largely  upon  the  number  of  slots  per  pole 
per  phase,  and  that  the  approximate  sine  wave  found  for  the 
space-value  of  the  flux  is  to  be  attributed  to  the  subdivision  of 
the  coils  of  each  pole  winding.  In  order  to  lay  emphasis  on  the 
fact  that  the  space  distribution  of  the  magnetism  is  largely 
independent  of  the  distribution  of  the  primary  coils,  and  to 
lend  simplicity  to  the  explanations,  the  initial  treatment  below 
will  be  based  on  the  assumption  of  coils  of  each  phase  being 
placed  in  a  single  slot  per  pole,  giving  the  maximum  of  con- 
centration. The  modifications  which  the  distribution  of  the 
coils  may  introduce  in  the  value  of  the  flux,  will  later  be  treated 
in  the  simplest  possible  manner.  It  is  well  at  this  point  to  make 


124  ALTERNATING  CURRENT  MOTORS. 

note  of  the  fact  that  the  effective  value  of  the  flux  threading  each 
primary  coil  is  determined  solely  by  the  primary  e.m.f,,  but  that 
the  space  distribution  of  the  flux  depends  upon  the  demands 
of  the  secondary  circuits. 

MAGNETIC  DISTRIBUTION  WITH  OPEN  SECONDARY. 

Figs.  57  to  77  indicate  the  behavior  of  the  magnetism  in  the 
separate  phase  windings,  and  throughout  the  air-gap  of  a  two- 
phase  induction  motor  both  when  the  secondary  is  on  open  cir- 
cuit and  when  it  is  on  closed  circuit  with  the  rotor  running  at 
synchronous  speed.  It  is  well  to  investigate  first  the  action 
which  takes  place  in  a  single  pole  winding  of  one  phase.  The 
effective  number  of  magnetic  lines  threading  this  winding  is 
determined  by  the  voltage  impressed  upon  the  coil  and  upon  the 
number  of  alternations  of  the  e.m.f.  At  each  instant  the  re- 
sultant magnetism  must  have  a  value  such  that  its  rate  of  change 
generates  in  the  coil  the  proper  counter  e.m.f.,  and  for  con- 
stant frequency  this  value  is  quite  independent  of  any  condition 
other  than  the  pressure  alone.  A  variation  in  the  reluctance 
of  the  path  of  the  lines  alters  only  the  magnetomotive  force 
necessary  to  produce  the  lines.  This  magnetomotive  force  is 
supplied  by  current  through  the  windings,  having  a  value  such 
that  the  ampere  turns  produce  the  required  magnetomotive 
force. 

If  the  individual  coils  of  the  second  phase  of  the  two-phase 
primary  winding,  which  are  located  on  the  core  90  electrical 
space  degrees  from  those  of  the  other  phase,  be  subjected  to  the 
same  conditions  as  assumed  for  the  first  phase,  obviously  the 
same  results  will  be  obtained.  If  the  coils  of  both  phases  be 
subjected  simultaneously  to  equal  effective  pressures  at  an  elec- 
trical-time displacement  of  90  degrees  from  each  other,  operat- 
ing conditions  for  a  two-phase  motor  will  be  obtained. 

At  each  instant  the  rate  of  change  of  interlinking  lines  and 
turns  per  pole  for  the  windings  of  each  phase  will  be  quite  in- 
dependent of  any  condition  other  than  the  instantaneous  value 
of  the  pressure  of  that  phase,  so  that  the  presence  of  the  wind- 
ings of  the  second  phase  can  in  no  manner  alter  the  effective 
value  of  interlinkages  of  the  first  phase  winding,  though  a  change 
in  distribution  of  actual  lines  may  occur. 

When  the  windings  of  one  phase  alone  are  energized,  the  core 


MAGNETIC  FIELD. 


125 


magnetism  evidently  reaches  its  zero  value  once  for  each  alter- 
nation of  the  pressure.  With  the  pressure  simultaneously  active 
on  the  second  phase  winding,  when  the  effective  magnetism 


Tta."'  " 


_. 


|Flui  in  -A"   _  [~~~     ~"| 

~lSp4_r_L 

L±:±~J  1 l 

T'~I     r~i — r~ 

j.j  t: 


.J       I       i -|_.J  i -L 

_.J  L._  L 


n 


FIGS.  .37-63. — Flux  De-     FIGS.  64-70. — Resultant  Figs.   71-77. — Result- 
manded  by  each  Phase;     Core  Flux;   Secondary       ant  Core  Flux;  Secorid- 
Secondary  open.  Open.  ary  Closed. 

which  threads  the  coils  of  the  first  phase  is  at  its  zero  value,  the 
magnetism  threading  the  other  phase  is  at  a  maximum.  Now, 
since  the  coils  of  both  phases  occupy  the  same  core  and,  in  fact, 


126  ALTERNATING  CURRENT  MOTORS. 

the  windings  of  each  phase  in  effect  surrourd  the  whole  polar 
area  of  the  core,  the  condition  of  zero  effective  magnetism  for 
one  phase  winding  can  be  accounted  for  only  by  the  fact  that 
each  pole  winding  of  that  phase  surrounds  an  equal  number  of 
north  and  of  south  lines.  See  Figs.  57  and  63. 

In  intermediate  conditions  of  the  magnetism,  the  number  of 
effective  lines  surrounded  by  the  windings  of  each  phase  depends 
upon  the  slope  of  the  e.m.f.  curve  of  each  phase  considered 
separately.  If  the  electromotive  force  of  each  phase  follows  a 
sine  curve  of  time- value,  the  relative  number  of  effective  lines 
surrounded  by  one  phase  will  vary  as  the  product  of  the  max- 
imum lines  into  the  cosine  of  the  angle  of  time,  while  those  of 
the  other  phase  at  the  same  instant  will  vary  as  the  sine  of  the 
same  angle. 

The  relative  changes  in  the  values  of  the  magnetism  threading 
each  coil  of  a  two-phase  motor  at  0,  15,  30,  45,  60,  75,  and  90 
time  degrees  (during  one-quarter  cycle)  are  shown  graphically 
in  Figs.  57,  58,  59,  60,  61,  62,  and  63,  respectively.  The  values 
indicated  are  those  which  would  be  found  for  each  primary  phase 
winding  operating  singly,  the  secondary  being  on  open  circuit. 
Figs.  64,  65,  66,  67,  68,  69,  and  70,  respectively,  show  the  re- 
sultant value  and  distribution  of  the  core  flux  when  the  two- 
phase  windings  are  subjected  simultaneously  to  electromotive 
forces  in  time  quadrature.  By  comparing,  say  Fig.  59  with 
Fig.  66,  it  will  be  seen  that  the  effective  value  of  the  flux  thread- 
ing each  coil  of  one  phase  is  in  no  way  altered  by  the  presence 
of  the  flux  demanded  by  the  e.m.f.  impressed  on  the  other  phase 
winding.  It  is  to  be  noted  particularly  that  the  flux  produced 
by  a  current  of  any  value  whatsoever  in  one-phase  winding  has 
absolutely  no  effect  upon  the  interlinkage  of  flux  with  the  coils 
of  the  other  phase.  It  is  noteworthy  also  in  this  connection 
that  the  magnetomotive  force  necessary  to  produce  a  certain 
effective  flux  in  one-phase  winding  is  neither  increased  nor  de- 
creased by  the  presence  of  the  flux  due  to  the  magnetomotive 
force  of  the  other  phase  winding.  That  is  to  say,  with  the 
secondary  on  open  circuit,  each  phase  winding  operates  as 
though  the  other  were  not  present,  and  as  though  it  alone 
occupied  the  core. 

A  comparison  of  Fig.  67  with  Figs.  64  and  70  and  the  inter- 
mediate Figs.  65,  66,  68,  and  69,  will  reveal  the  fact  that, 


MAGNETIC  FIELD.  127 

when  the  secondary  is  on  open  circuit,  the  resultant  core  flux 
changes  in  mechanical  position  along  the  core,  it  varies  in  space 
distribution,  and  alters  both  in  magnetic  density  and  in  total 
existing  magnetic  lines.  Thus  in  Fig.  67,  the  flux  is  distributed 
over  only  one-half  of  the  core  area,  where  the  magnetic  density 
is  \/2  times  that  indicated  in  Figs.  64  and  70,  and  the  total 
number  of  magnetic  lines  is  only  \/J5  times  that  shown  in  either 
Fig.  64  or  Fig.  70.  It  is  seen,  therefore,  that  under  the  condition 
here  assumed,  giving  separately  to  the  density  and  to  the  total 
flux  in  Fig.  67  or  Fig.  70,  the  arbitrary  value  100,  the  density 
of  the  core  flux  varies  from  100  to  141.4  and  the  total  number 
of  lines  existing  on  the  core  varies  from  100  to  70.7  four  times 
during  each  cycle. 

By  noting  the  electrical  space  positions  of  the  resultant  core 
flux  in  Figs.  64,  67,  and  70,  it  will  be  seen  that  the  flux  moves 
along  the  air-gap  45  electrical  space  degrees  during  45  electrical 
time  degrees,  and  that  it  advances  a  total  of  90  electrical  space 
degrees  during  90  electrical  time  degrees.  Thus,  even  when  the 
secondary  is  on  open  circuit  it  travels  around  the  air-gap  at  a 
certain  definite  speed  termed  "  synchronous,"  although  it 
varies  in  value  and  in  space  distribution. 

MAGNETIC  DISTRIBUTION  WITH  CLOSED  SECONDARY. 

Assume,  now,  that  the  rotor  is  driven  by  some  external  means 
so  that  it  travels  at  exactly  synchronous  speed.  If  the  secondary 
conductors  are  on  open  circuit,  electromotive  forces  will  be  gen- 
erated in  them  locally  by  the  rate  of  change  of  the  synchron- 
ously moving  core  flux.  If  now  the  secondary  circuits  be  closed 
the  electromotive  forces  generated  by  the  changing  core  flux 
will  produce  currents  in  the  secondary,  which  tend  to  prevent 
the  variation  in  the  magnetism  which  links  with  the  secondary 
conductors. 

If  the  secondary  circuits  are  thoroughly  distributed  over  the 
secondary  core,  and  are  of  perfect  conductivity,  then  the  most 
minute  change  in  the  value  or  space  distribution  of  the  synchron- 
ously moving  magnetism  will  produce  an  enormous  secondary 
current  tending  to  maintain  both  the  distribution  and  the  value 
of  the  flux.  As  stated  previously,  the  e.m.f.  across  each  primary 
coil  demands  that  a  certain  effective  value  of  core  flux  at  each 
instant  threads  through  that  coil,  but  the  requirements  of  the 


128  ALTERNATING  CURRENT  MOTORS, 

e.m.f.  are  met  as  fully  with  one  space  distribution  of  the  flux 
as  with  another,  so  long  as  the  effective  value  remains  the  same- 

The  exacting  requirements  of  the  secondary  conductors  and  of  the 
electromotive  forces  impressed  upon  the  phase  windings  of  the  pri- 
mary coils  are  completely  fulfilled  when  the  core  ftux  assumes  a  sine 
curve  of  electrical  space  distribution,  as  shown  in  Figs.  71  to  77, 
inclusive.  The  proof  of  this  statement  is  given  below. 

A  comparison  of  the  sine  curve  of  Fig.  71  with  the  rectangular 
curve  of  Fig.  64  will  serve  to  determine  the  value  of  the  max- 
imum ordinate  of  the  sine  curve.  Since  the  area  included  be- 
tween each  curve  and  its  base  line  must  be  the  same  in  the 
two  cases,  the  maximum  ordinate  of  the  sine  curve  must  be 

—  times  the  maximum  ordinate  in  the  curve  of  Fig.  64,  as  will 

be  verified  incidentally  below. 

A  comparison  of  Fig.  67  with  Fig.  64  will  show  at  a  glance 
that  the  area  enclosed  by  the  former  curve  is  much  smaller  than 
that  enclosed  by  the  latter,  and  the  question  naturally  arises  as 
to  the  possibility  of  any  curve  which  surrounds  a  constant  area 
serving  to  meet  the  demands  for  effective  areas  which  differ  so 
widely  as  do  those  indicated  by  the  curves  of  Figs.  64  and  67. 

It  will  be  observed  that  in  connection  with  the  sine  curves  of 
constant  value  but  variable  position  shown  in  Figs.  71  to  77, 
inclusive,  there  are  given  also  rectangular  curves  of  constant 
position  but  variable  value,  which,  as  will  be  seen  from  Figs. 
57  to  63,  inclusive,  indicate  the  demand  for  effective  area 
(magnetism)  made  by  the  e.m.f.  of  phase  A.  It  remains  now 
to  be  demonstrated  that  between  the  two  points  M  and  N 
which  are  constant  in  position,  there  is  intercepted  from  the 
area  represented  by  the  sine  curve  as  it  moves  along  synchron- 
ously, an  effective  area  equal  in  magnitude  at  all  times  to  the 
area  represented  by  the  rectangular  curve. 

Referring  now  to  Fig.  71  let 

h  =  maximum  ordinate  of  sine  curve, 
then 

y  =  h  sin  x  is  the  equation  of  the  sine  curve. 

The  average  ordinate  of  the  whole  curve  from  M  to  N  (n  ra- 
dians) is 

* 

(1) 


sin  x  d  x  =  —       —  cos  x  =  - 


MAGNETIC  FIELD.  129 

Since  the  area  of  the  sine  curve  is  equal  to  that  of  the  rectan- 
gular curve  in  Fig.  71,  the  maximum  ordinate  of  the  sine  curve 

is  equal  to  —  times  that  of  the  rectangular  curve,  as  mentioned 

previously . 

In  Fig.  74,  let  the  distance  that  the  zero  ordinates  of  the 
curve  have  traveled  from  M  and  from  N  be  represented  by  a. 
Then  the  area  below  the  base  line  will  have  the  numerical  value 

of 

[a 
—   COS  X 
n 


-J: 


h  [  —  cos  a  —  ( —  cos  0)]  =  h  (1  —  cos  a)  (2) 

This  area  will  in  effect  neutralize  an  equal  positive  area,  so  that 
the  remaining  effective  area  will  be 

h    I       sin  x  d  x  —  2  h  sin  x  d  x  = 


C~  Ca 

I       sin  x  d  x  —  2  h 
Jo  Jo 


2  h- 2  h  (I -cos  a)  =  2  h  cos  a  (3) 

This  equation  shows  that  the  effective  area  bounded  between 
the  ordinates  at  M  and  A7  and  the  sine  curve  is  proportional 
directly  to  the  cosine  of  the  angle  of  displacement  of  the  curve 
from  the  position  giving  the  maximum  area,  as  was  assumed 
above. 

The  interpretation  of  the  above  equation  is  that  the  effective 
area  bounded  by  the  sine  curve  and  the  ordinates  M  and  N  in 
Figs.  71  to  77  inclusive  is  in  each  case  equal  to  the  area  repre- 
sented by  the  indicated  rectangular  curve.  That  is  to  say, 
magnetism  of  constant  magnitude  and  distributed  according  to  a 
sine  curve  of  electrical  space  value  will,  when  traveling  syn- 
chronously around  the  air-gap,  generate  within  each  coil  an 
electromotive  force  having  a  sine  wave  of  electrical-time  value. 
The  relative  electrical-time-phase  position  of  the  electromotive 
forces  of  coils  distributed  around  the  air-gap  will  depend  solely 
upon  the  electrical-space  positions  of  the  coils.  If  independent 
sets  of  coils  are  located  at  intervals  of  90  electrical-space  degrees, 
the  electromotive  forces  generated  therein  will  vary  from  each 
other  by  \  period,  and  the  coils  may  be  interconnected  to  form 


130  ALTERNATING  CURRENT  MOTORS. 

a  four-phase  motor,  or  the  opposite  phase  windings  of  this  four- 
phase  machine  may  be  joined  so  as  to  form  the  familiar  two- 
phase  induction  motor.  Similarly,  if  independent  sets  of  coils 
are  located  at  intervals  of  60  electrical-space  degrees,  the  elec- 
tromotive forces  generated  therein  will  vary  from  each  other 
by  J  period  and  the  coils  may  be  interconnected  to  form  a  six- 
phase  motor,  or  the  opposite  phase  windings  of  this  six-phase 
machine  may  be  joined  so  as  to  form  the  familiar  three-phase 
induction  motor. 

DETERMINATION  OF  CORE  FLUX. 

The  determination  of  the  value  of  the  core  flux  can  be  based 
either  upon  the  fundamental  transformer  equation  or  upon  the 
equation  used  with  alternating-current  generators.  Let 

n  =  number  of  turns  in  series  per  primary  coil. 

E  =  effective  value  of  primary  e.m.f.  per  coil. 
/  =  frequency  in  cycles  per  second. 

From  transformer  relations 


(4) 


where  (/>m  is  the  maximum  value  of  the  total  flux  threading  the 
coil. 

Let  A  =  the  total  air-gap  area  covered  by  the  coil. 

When  the  secondary  is  on  open  circuit,  and  only  one  phase 
winding  is  active, 

„      _  *.   .         WE 

~A     ~  V2  xfnA 


where  Bm  is  the  maximum  magnetic  density  at  any  point  along 
the  air-gap.     (See  Fig.  57.) 

When  the  secondary  is  on  open  circuit  and  both  phase  wind- 
ings are  active.     (See  Fig.  67.) 


B         -m- 

-A-     ~xfnA 

When  the  secondary  circuit  is  completely  closed  and  the  rotor 
is  running  at  synchronous  speed.      (See  Figs.  71  to  77.) 

_*_£„,  _        IV  E 

2A     ~  2~nA 


MAGNETIC  FIELD.  131 

Treating  the  machine  now  as  an  alternator  having  n  turns  in 
series  on  the  armature,  with  a  flux  of  <f>m  total  lines  per  pole, 

fn^m  (8) 


and  the  maximum  magnetic  density  is,  as  found  above, 

_*<£._        10»£ 
"  2  A     ~ 


The  proof  of  the  identity  of  the  equations  derived  from  trans- 
former and  from  alternator  relations  as  given  above,  has  been 
based  upon  the  assumption  of  sine  curves  of  electromotive 
forces.  It  is  evident  that,  since  the  effective  magnetism  thread- 
ing each  coil  must  vary  at  each  instant  according  to  the  instan- 
taneous value  of  the  e.m.f.,  when  the  e.m.f.  wave  is  distorted  the 
core  flux  must  likewise  vary  from  a  sine  curve  of  electrical  space 
distribution.  It  is  an  interesting  conclusion,  which  permits  of 
easy  verification  that  the  mechanical  distribution  of  the  core  flux 
follows  a  wave  of  electrical-space  value  similar  in  all  respects  to 
the  electrical-time  value  of  the  primary  electromotive  force.  This 
fact  will  be  appreciated  immediately  if  one  considers  that  the 
instantaneous  value  of  the  e.m.f.  generated  in  each  armature 
conductor  depends  directly  on  the  local  magnetic  density  of  the 
field  through  which  it  is  moving  at  that  instant. 

EFFECT  ON  CORE  FLUX  OF  USING  DISTRIBUTED  WINDING. 

The  problem  of  determining  the  effect  of  distributing  the 
windings  of  each  phase  over  a  certain  portion  of  the  air-gap 
instead  of  concentrating  them  in  one  slot  per  pole,  as  assumed 
above,  is  rendered  extremely  simple  by  treating  the  machine  as 
an  alternator,  as  was  intimated  in  the  opening  paragraphs  of 
this  chapter.  If  the  conductors  which  cross  the  face  of  the  core 
and  are  joined  in  series  to  form  a  primary  coil  of  one  phase,  are 
distributed  over  /?  electrical-space  degrees,  then  the  resultant 
e.m.f.  for  a  certain  core  magnetism  is  less  than  the  arithmetical 
sum  of  the  individual  e.m.f.  of  the  several  conductors  in  the 
ratio  of  the  cord  of  angle  ft  to  the  arc  of  the  same  angle.  This 
result  follows  directly  from  the  fact  that  the  individual  e.m.fs. 
are  not  in  phase  one  with  the  other  and  it  is  necessary  to  take 
the  vector  sum  of  them. 


132  ALTERNATING  CURRENT  MOTORS. 

In  a  two-phase  motor   (the  equivalent  of  a  four-phase  ma- 
chine) the  angle  j)  is 

angle  /?  =  90  degrees 
arc  of  {3  =  '— 

cord  of  /?  =  \/2~ 

Therefore,  in  a  two-phase  motor,  the  maximum  magnetic  den- 
sity may  be  expressed  as 

B  *  W*E         -*     WE 

~  'nA  "  8'  fnA 


In  a  three-phase  motor  (the  equivalent  of  a  six-phase  machine) 
the  angle  /?  is 

angle  ^9  =  60  degrees 

arc  of  0  =  | 

d 

cord  of  /?  =  1 

Therefore,  in  a  three-phase  motor  the  maximum  magnetic  den- 
sity may  be  expressed  as 

*          10*  £  *         lO^E 

"  3  '  2  x/2  /  n  A   ~    6  x/2     /  »  ^ 

Both  equation  (10)  and  equation  (11)  have  been  derived  on 
the  basis  of  the  initial  assumption  that  each  turn  of  each  coil 
spans  an  arc  of  180  electrical  space  degrees.  It  is  evident  that 
if  each  turn  covers  an  area  less  than  that  indicated  by  an  arc 
of  180  degrees,  the  magnetic  density  must  have  a  value  greater 
than  that  given  by  these  equations.  In  commercial  induction 
motors  one  side  of  each  coil  is  placed  in  the  bottom  of  a  certain 
slot  and  the  return  side  of  the  same  coil  is  placed  in  the  top  of 
another  slot,  with  an  arc  of  less  than  180  electrical  space  degrees 
between  the  slots. 

Let  the  span  of  each  coil  be  7-  electrical-space  degrees,  then 
the  e.m.f.  generated  in  the  one  side  of  each  coil  will  be  f  elec- 
trical-time degrees  out  of  phase  with  the  e.m.f.  in  the  other 
side  of  the  same  coil.  If  e  is  the  e.m.f.  in  one  side  of  a  coil, 
the  resultant  e.m.f.  of  the  coil  will  be 

EC  =  V2  e  Vl+cos(180-r)  (12) 


MAGNETIC  FIELD.  133 

When  f  =  180  equation   (12)  reduces  to 

Ec  =  2e  (13) 

Hence,  in  general,  for  a  two-phase  motor 


TT     10*  E     x/l-fcosqSO-r) 

8  •  rA  •  -    ~ 


and  for  a  three-phase  motor, 
R       .  JL     WE 


+  cos 


The  last  two  equations  refer  to  the  magnetic  density  imme- 
diately at  the  bottom  of  the  teeth  of  the  primary  core.  The 
local  magnetic  density  in  the  air-gap  will  depend  upon  the 
relative  size  of  the  slots  and  the  teeth,  and  will  be  greater  than 
that  shown  by  these  equations. 

EFFECT  ON  CAPACITY  OF  VARYING  THE  GROUPING  OF  COILS. 

An  examination  of  the  formation  of  equation  (10),  (11),  (14), 
and  (15)  will  reveal  the  fact  that  if  in  a  certain  induction  motor 
the  maximum  value  of  the  magnetic  density  is  to  remain  con- 
stant while  the  coils  are  interconnected  in  different  ways,  the 
e.m.f.  of  each  group  of  coils  may  be  represented  relatively  as 
the  cord  of  the  arc  in  electrical  space  degrees  which  is  covered 
by  the  coils  in  the  group.  It  follows  therefore  that  if  one-half 
of  the  coils  are  connected  continuously  in  series  the  total  e.m.f. 
of  the  group  of  n  coils  in  each  of  which  there  is  an  e.m.f.  of  e 

volts  will  be  -  -  e  volts.     If   this   value   of   volts  be   taken  as 

7T 

unity,  for  the  sake  of  comparison,  then  when  one-third  of  the 
coils  are  joined  in  continuous  series  the  total  voltage  of  the  group 
will  be  .866  volts.  Likewise  a  group  containing  one-fourth  of 
the  coils  would  have  a  voltage  of  .707,  and  a  group  containing 
one-sixth  of  the  coils  would  have  a  voltage  of  .500. 

If  now  it  be  assumed  that  each  coil  is  to  carry  the  same 
current  as  the  other  coils,  then  the  volt-amperes  per  group 
will  vary  directly  with  the  voltage.  In  consequence  of  this 
fact  the  total  volt-amperes  of  an  induction  motor  when  oper- 
ated at  constant  maximum  magnetic  density  in  the  core  and 


per  Group. 

Volt-amperes. 

Machine. 

1.000 

2.000 

Single-phase 

.866 

2.598 

Three-phase 

.707 

2.828 

Quarter-phase 

.500 

3.000 

Six-phase 

(Three-phase) 

134  ALTERNATING  CURRENT  MOTORS. 

constant  current  density  in  the  coils  will  be  as  follows  for  various 
groupings  of  the  coils,  assuming  unit  current: 

Number  of  Voltage  Total  Type  of 

Groups. 
2 
3 
4 
6 


The  relations  shown  in  the  above  table  were  commented  on 
in  the  preceding  chapter.  It  will  be  noted  that  the  volt-amperes 
rating  of  a  single-phase  motor  are  .707  times  that  of  a  quarter- 
phase  machine.  A  little  study  will  show  that  a  few  of  the  pri- 
mary coils  of  each  group  may  be  removed  without  seriously 
decreasing  the  volt-amperes  of  the  single-phase  machine.  Thus 
it  is  possible  to  materially  decrease  the  primary  copper  without 
a  proportionate  decrease  in  the  volt-amperes,  as  shown  by  the 
following  table: 

Percentage  of  Percentage  of  Saving  in  Decrease  in 

Coils.  Volt-amperes  Copper  Input 

100.00  100.000                      .00  .00 

88.89  98.48  11.11,  1.52 

77.78  93.97  22.22  6.03 

66.67  86.60  33.33  13.40 

55.56  76.60  44.44  23.40 

50.00  70.70  50.00  29.30 

It  is  seen  from  the  above  that  a  portion  of  the  primary  copper 
could  be  removed  and  yet  the  performance  of  the  machine  would 
be  only  slightly  affected.  It  might  seem  that  this  fact  would 
permit  of  a  considerable  saving  in  material,  but  a  motor  thus 
constructed  would  not  in  general  be  capable  of  being  rendered 
self-starting  from  its  primary  circuits.  It  is  the  usual  practice 
therefore  to  wind  the  primary  completely  and  to  use  only  a  por- 
tion of  the  coils  during  normal  operation,  all  of  the  coils  being 
employed  during  the  starting  period.  Commercial  single-phase 
induction  motors  are  frequently  constructed  as  uniformly-wound" 
three-phase  machines,  or  as  unsymmetrically  wound  two-phase 
machines.  In  the  latter  case  the  "  main  "  winding  contains 


MAGNETIC  FIELD 


135 


about  twice  as  much  copper  as  the  "  starting  "  winding,  and  it 
occupies  two-thirds  of  the  core  slots. 

EXCITING  WATTS  IN  INDUCTION  MOTORS. 

In  the  treatment  above,  the  part  played  by  the  primary  cur- 
rent in  producing  the  revolving  field  has  been  practically  neg- 
lected. It  is  well  in  this  connection  to  show  how  the  value  of 
the  exciting  current  may  be  determined  directly  from  the 
volume  of  the  air-gap  and  the  volume  of  the  core  material, 
without  reference  to  the  required  magnetomotive  force,  the 
number  or  the  distribution  of  the  primary  coils. 

In  Fig.  78a,  let 

A  =  area  of  magnetic  path,  in  sq.  cm. 
/  =  length  of  path  in  iron,  in  cm. 


-4- 


FIG.  78A.  —  Simple  Magnetic  Circuit. 

permeability  of  iron. 

length  of  path  in  air. 

number  of  turns  of  coil. 

effective  value  of  impressed  e.m.f.,  in  volts. 

any  chosen  value  of  flux. 

any  chosen  value  of  exciting  current,  in  amperes. 

effective  value  of  exciting  current. 
$m  =  maximum  value  of  flux. 
From  fundamental  magnetic  relations 


d  = 

n  = 

E  = 

<j>  = 

i  — 

Iq  = 


Flux 


n 


m.m.f. 
reluctance 


(16) 


136  ALTERNATING  CURRENT  MOTORS 

As  is  well  known,  the  reluctance  of  commercial  magnetic 
material  is  not  constant  for  all  densities,  and  hence  it  is  not 
proper  to  assume  that  the  exciting  current  is  sinusoidal  when 
the  flux  is  sinusoidal.  When  iron  is  included  in  the  magnetic 
path,  the  exciting  current  wave  will  be  peaked.  When  the 
major  portion  of  the  reluctance  of  the  path  is  in  air,  the  effect 
of  the  distortion  produced  by  the  presence  of  the  variable  re- 
luctance of  the  iron  will  not  in  general  be  very  marked,  and 
for  all  practical  purposes  it  may  well  be  neglected. 

Thus,  if  the  maximum  value  of  the  exciting  current  is  im, 
the  effective  value  will  be  slightly  different  from  \/.5  *w»  but 
since,  in  any  event,  the  actual  value  of  im  cannot  be  predeter- 
mined with  a  high  degree  of  accuracy,  due  to  the  fact  that  the 
true  value  of  /£  is  not  known,  it  is  safe  to  assume  that  for  in- 
duction motors  no  measurable  error  is  introduced  by  representing 
the  effective  value  of  the  exciting  current  by  the  equation. 

:       /«   =    V^5«m  (17) 

1     4  n  !   ,_  T  A_ 

(18) 


id  -fUi 


IQ=     ~ 
But 

E  =  V-2^n}<t,m  '  '  (2Q) 

Hence  the  (quadrature)  exciting  watts  may  be  expressed  as 


Let  V i  =  A  I   =  volume  of  iron. 

Va  =  A  d  =  volume  of  air. 
Then 

A+T) ?•***. vai  pi  (22) 


(23) 


MAGNETIC  FIELD. 


137 


where  Bm  is  the  maximum  magnetic  density,  in  lines  per  square 
centimeter. 

The  interpretation  of  equation  (23)  is,  that  in  order  to  ascer- 
tain the  (quadrature)  exciting  watts  it  is  necessary  to  know 
only  the  maximum  magnetic  density,  the  volume  and  the  per- 
meability of  the  iron,  and  the  volume  of  the  air-gap.  That  is 
to  say,  the  very  quantities  which  are  necessary  in  order  to  deter- 
mine the  core  losses,  will  serve  simultaneously  for  the  determination 
of  the  (quadrature)  exciting  watts,  when  the  permeability  of  the 
core  is  known. 

Although  the  erratic  behavior  of  iron  with  reference  to  the 
change  in  its  permeability  cannot  be  reduced  to  a  mathematical 
expression,  it  is  found  that  for  most  practical  purposes  the  per- 
meability of  the  iron  used  in  transformers  and  induction  motors 


FIG.  78s. — Composite  Magnetic  Circuit. 

may  be  expressed  with  a  fair  degree  of  accuracy,  throughout  the 
range  of  density  from  Bm  =  0  to  Bm  =  15,000,  by  the  equation 


/£  =  2,800-3.2 


(24) 


Although  the  proof  given  above  for  equation  (23)  refers  pri- 
marily to  the  magnetic  circuits  represented  in  Fig.  78a,  it  can  be 
shown  that  the  facts  stated  in  connection  with  equation  (23) 
apply  equally  as  well  to  the  circuits  indicated  in  Fig.  78b  and  to 
the  more  complex  circuits  existing  in  both  single-phase  and 
polyphase  motors. 

Thus,  making  use  of  proper  subscripts,  in  Fig.  78b, 


W, 


9  5  f  f~  /  F"  •' 

To5"  lB2im  (Via  +  77 


(25) 


138  ALTERNATING  CURRENT  MOTORS. 

It  may  be  shown  theoretically  and  verified  experimentally, 
that  the  (quadrature)  exciting  watts  of  a  certain  polyphase 
motor  are  the  same  in  value  when  all  phase  windings  are  used  or 
when  only  one  winding  is  subjected  to  the  primary  pressure. 
Thus,  when  one  phase  winding  of  a  two-phase  motor  is  open- 
circuited,  the  other  winding  immediately  takes  double  its  former 
value  of  (quadrature)  exciting  current,  the  (quadrature)  excit- 
ing watts  remaining  the  same.  These  facts  are  discussed  more 
fully  in  the  next  chapter. 

It  seems,  therefore,  that  the  most  logical  way  to  determine 
the  exciting  current  of  an  induction  motor  is  to  ascertain  the 
density  of  the  magnetism  in  the  core  and  in  the  air-gap,  and  then 
calculate  the  quadrature  watts,  just  as  one  ordinarily  calculates 
the  core  loss  watts. 

MAGNETIC    FIELD    IN    THE    SINGLE-PHASE    INDUCTION   MOTOR. 

As  was  mentioned  in  a  previous  chapter,  when  the  circuits 
•of  a  polyphase  induction  motor  operating  near  synchronism 
.are  so  arranged  as  to  convert  the  machine  into  a  single-phase 
motor,  the  revolving  field,  which  was  previously  due  to  the 
•combined  actions  of  certain  components  of  the  displaced  poly- 
phase currents,  continues  to  exist,  and  the  action  of  the  machine 
in  developing  mechanical  power  is  subjected  to  almost  no  change. 
It  is  equally  well  known  that  the  quadrature  "  speed  "  com- 
ponent of  the  magnetic  field  is  produced  by  current  in  the  sec- 
ondary, and  that  the  magnetomotive  force  represented  by  this 
secondary  current  must  be  supplied  by  a  component  of  pri- 
mary current.  On  account  of  the  fact  that  the  secondary 
current  which  produces  the  quadrature  "  speed  field"  occupies 
a  position  in  space  such  that  it  cannot  possibly  react  directly 
on  the  field  produced  by  the  primary  current,  it  is  not  immedi- 
ately apparent  in  what  manner  the  "  quadrature  watts  "  for 
the  "  speed  field  "  are  supplied  by  the  primary  exciting  mag- 
netomotive force. 

A  popular  method  of  treating  the  internal  behavior  of  a 
single-phase  induction  motor  is  the  one  due  to  Ferraris,  who 
showed  that  the  simple  alternating' field  can,  in  all  of  its  effects, 
be  replaced  by  two  revolving  fields  moving  in  opposite  directions, 
the  maximum  value  of  each  being  equal  to  one-half  of  the  maxi- 
mum value  of  the  alternating  field.  By  means  of  this  method 


MAGNETIC  FIELD.  139 

it  is  possible  to  ascertain  the  distribution  of  current  in  the 
secondary,  and  the  reaction  of  certain  components  of  the  sec- 
ondary current  upon  the  primary,  but  the  present  writer  be- 
lieves that  the  actual  significance  of  the  results  obtained,  as 
viewed  by  the  average  reader,  are  greatly  obscured  by  the 
difficulty  in  distinguishing  the  imaginary  from  the  real  when, 
the  two  are  so  closely  interwoven. 

A  prominent  writer  of  the  present  day  states  that  "  the 
cause  of  the  cross  magnetization  in  the  single-phase  induction 
motor  near  synchronism,  is  that  the  induced  armature  currents 
lag  90°  behind  the  inducing  magnetism  and  are  carried  by  the 
synchronous  rotation  90°  in  space  before  reaching  their  max- 
imum ";  and  that  "below  synchronism  the  induced  armature 
currents  are  carried  less  than  90°,  and  thus  the  cross  magneti- 
zation due  to  them  is  correspondingly  reduced  and  becomes 
zero  at  standstill."  It  is  greatly  to  be  doubted  if  these  state- 
ments convey  any  physical  idea  whatever  to  a  mind  not  already 
thoroughly  familiar  with  the  facts. 

Although  the  method  outlined  below  will  not  serve  to  present 
any  facts  which  cannot  be  ascertained  by  other  methods  which 
have  frequently  been  employed,  yet  it  is  believed  that 
much  good  can  be  accomplished  by  drawing  attention  to  the 
fact  that  all  of  the  phenomena  connected  with  the  production 
of  the  magnetic  field  in  the  single-phase  induction  motor  can 
be  investigated  with  the  utmost  simplicity  by  dealing  directly 
with  well-known  electro-magnetic  relation  without  resorting 
to  imaginary  physical  or  mathematical  representations.  This 
method  has  been  touched  upon  in  a  preceding  chapter.  It  is 
believed  that  a  more  extended  discussion  thereof  is  desirable, 
at  this  point. 

PRODUCTION  OF  SPEED-FIELD  CURRENT. 

Fig.  79  shows  a  two-pole  model  of  a  single-phase  induction 
motor  which  is  represented  as  possessing  four  mechanical  poles, 
two  of  which  (1  and  3)  are  excited  by  single-phase  alternating 
current.  Merely  for  sake  of  simplicity  in  explanation,  mechanical 
poles  2  and  4  are  indicated  as  subjected  exclusively  to  the  flux 
of  the  "  speed  field."  Under  any  condition  of  operation  the 
flux  in  poles  1  and  3  is  determined  directly  by  the  primary 
e.m.f.,  modified  by  the  volts  consumed  in  the  local  impedance 


140 


ALTERNATING  CURRENT  MOTORS. 


of  the  primary  coil  by  the  current  which  flows  therethrough. 
That  is  to  say,  the  flux  in  these  poles  follows  the  laws  which 
relate  to  stationary  transformers;  no  action  which  takes  place 
in  the  secondary  can  rob  the  primary  of  this  transformer  fea- 
ture. In  dealing  with  the  secondary,  however,  it  is  necessary 
to  recognize  the  fact  that  each  rotor  conductor  is  subjected 
to  four  distinct  electromotive  forces — the  e.m.f.'s  produced  by 
the  rate  of  change  of  the  transformer  and  of  the  speed  fields, 
and  the  e.m.f's  generated  by  the  motion  of  the  rotor  through 
the  transformer  and  speed  fields.  Each  of  these  e.m.f's  will 


Primary 
Current 


Transformer 

>        Primary 

Field 

S 

>        Current 

3 

FIG.  79.— Production  of  "  Speed-field  "  Current. 

be  treated  separately   and  the  combined  effects  will  then  be 
investigated. 

When  the  rotor  is  moving  across  the  transformer  field  in 
the  direction  indicated,  there  will  be  generated  in  each  of  the 
conductors  under  the  poles  an  e.m.f.  proportional  to  the  pro- 
duct of  the  field  magnetism  and  the  speed  of  the  rotor.  Evi- 
dently if  the  speed  be  constant,  of  whatsoever  value,  this  e.m.f. 
will  vary  directly  with  the  strength  of  magnetism;  that  is, 
will  be  maximum  when  the  magnetism  is  maximum,  and 
zero  at  zero  magnetism.  Qther  conditions  remaining  the  same 
the  maximum  value  of  this  secondary  e.m.f.  will  vary  directly 
with  the  speed  of  the  rotor. 


MAGNETIC  FIELD.  141 

If  the  circuits  of  the  rotor  conductors  be  closed,  there  will 
tend  to  flow  therein  currents  of  strengths  depending  directly 
upon  the  e.m.f.'s  generated  in  the  conductors  at  that  instant 
and  inversely  upon  the  impedance  of  the  rotor  conductors. 
The  current  which  flows  through  the  rotor  circuits  at  once 
produces  a  magnetic  flux  which  by  its  rate  of  change  in  value 
generates  in  the  rotor  conductors  a  counter  e.m.f.  opposing  the 
e.m.f.  that  causes  the  current  to  flow,  and  of  such  a  value 
that  the  difference  between  it  and  this  e.m.f.  is  just  sufficient 
to  cause  to  flow  through  the  local  impedance  of  the  conductors 
a  current  whose  magnetomotive  force  equals  that  necessary 
to  drive  the  required  lines  of  magnetism  through  the  reluctance 
of  their  paths.  Since  this  latter  magnetism  must  have  a  rate  of 
change  equal  (approximately)  to  the  e.m.f.  generated  in  the  rotor 
conductors  by  their  motion  across  the  primary  field,  and  since 
this  e.m.f.  is  in  time  phase  with  the  primary  field,  it  follows 
that  this  magnetism  must  have  a  value  proportional  to  the  rate 
of  change  of  the  primary  magnetism;  that  is,  it  is  (approxi- 
mately) in  time  quadrature  to  the  primary  magnetism. 

A  study  of  the  direction  of  the  currents  in  the  rotor  under  the 
conditions  assumed  will  show  that  when  a  north  pole  at  1  (in 
Fig.  79)  has  reached  its  maximum  value  and  is  decreasing  to- 
wards zero,  the  speed  field  is  building  up  with  a  north  pole  at 
2,  and  that  this  pole  continues  to  increase  in  strength  until  the 
magnetism  at  1  reverses  its  direction.  Thus,  it  may  be  stated 
that  the  north  pole  of  the  resultant  magnetism  travels  in  the 
direction  of  motion  of  the  rotor.  Since  the  rapidity  of  reversal 
in  sign  of  the  "  transformer  field  "  poles  and  of  the  "  speed  field  " 
poles  depends  solely  upon  the  frequency,  it  may  be  stated  that 
the  resultant  field  revolves  at  synchronous  speed.  The  "  speed 
field  "  is  equal  (approximately)  to  the  product  of  the  "  trans- 
former field  "  and  the  speed,  with  synchronism  as  unity.  Thus 
the  resultant  field  is  at  any  speed  elliptical  as  to  electrical  space 
representation;  one  axis  of  the  ellipse  is  determined  by  the 
"  transformer  field,"  while  the  other  depends  upon  the  speed. 
At  synchronism  the  ellipse  becomes  a  circle;  above  synchronism 
the  ellipse  has  its  major  axis  along  the  "  speed  field  ";  at  zero 
speed  the  ellipse  is  a  straight  line,  which  means  that  at  standstill 
there  is  no  "  space  "  quadrature  flux  and  hence  no  revolving 
field. 


142 


ALTERNATING  CURRENT  MOTORS. 


Reviewing  the  electromagnetic  processes  just  discussed,  it 
will  be  noted  that  the  e.m.f.  which  produces  the  "  speed  field  " 
current  is  caused  by  the  motion  of  the  rotor  through  the  "  trans- 
former field  "  and  is  opposed  by  the  rate  of  change  of  the  "  speed 
field  "  through  the  rotor  circuits.  The  mechanical  position  of 
the  "  speed  field  "  current  with  reference  to  the  primary  coil 
prevents  it  from  reacting  directly  on  the  "  transformer  field." 
It  remains  to  investigate  the  effect  of  the  e.m.f's  generated  in 
the  secondary  by  the  rate  of  change  of  the  "  transformer  field  " 
through  the  rotor  conductors  and  by  the  motion  of  the  rotor 
conductors  through  the  "  speed  field."  It  will  be  noted  at  once 


N1 

Transformer 

)        Primary 

Field 

1         Current 

x^^TT^s 

* 

FIGS.  80A  and  80B. — Production  of  Transformer  Secondary 
Currents  and  Electromotive  Forces. 


that  a  current  due  to  either  of  these  e.m.f's  would  be  in  position 
to  tend  to  affect  the  "  transformer  field." 

TRANSFORMER  FEATURES  OF  THE  SINGLE-PHASE  INDUCTION 

MOTOR. 

Referring  now  to  Fig.  80a  assume  initially  for  sake  of  simplicity 
that  the  rotor  revolves  at  absolutely  synchronous  speed  (being 
driven  by  some  external  means).  As  noted  above,  the  trans- 
former e.m.f.  of  the  "  speed  field  "  in  the  rotor  is  slightly  less 
than  the  speed  e.m.f.  of  the  "  transformer  field  ,"  and  is  out  of 
time  phase  therewith,  by  an  amount  equal  to  the  e.m.f.  necessary 


MAGNETIC  FIELD.  143 

to  cause  the  "  speed  field"  current  to  flow  through  the  "  local  " 
impedance  of  the  rotor  conductors.  It  is  seen  at  once,  there- 
fore, that  the  speed  e.m.f.  of  the  "  speed  field"  in  the  rotor 
differs  from  the  transformer  e.m.f.  of  the  "  transformer  field  " 
by  an  exactly  equal  amount,,  so  that  a  current  exactly  equal 
to  the  "  speed  field  "  current  is  produced  in  the  rotor  in  an 
electrical  space  position  such  that  its  magnetomotive  force 
tends  directly  to  affect  the  "  transformer  field."  Since  the 
"  transformer  field  "  must  have  the  value  demanded  by  the 
primary  e.m.f.,  a  current  equal  in  magnetomotive  force  and 
opposite  in  direction  to  this  component  of  the  secondary  current 
must  flow  in  the  primary  coil.  As  indicated  in  Fig.  80a,  and 
as  may  be  verified  by  a  study  of  the  fluxes  and  currents,  this 
component  of  the  secondary  current  has  a  time  phase  position 
to  tend  to  decrease  the  "  transformer  field,"  so  that  the  opposing 
current  in  the  primary  appears  as  an  added  component  of  the 
primary  exciting  current.  Thus  the  "  speed  field  "  current  is 
accurately  represented  in  the  exciting  magnetomotive  force 
supplied  by  the  primary  current. 

It  is  interesting  to  note  that  the  "  added  "  component  of  the 
primary  exciting  current  depends  upon  the  reluctance  of  the 
path  taken  by  the  flux  of  the  "speed  field";  when  the  air 
gap  traversed  by  the  "  speed  field  "  is  much  greater  than  that 
through  which  the  "  transformer  field  "  passes  (as  shown  in 
Figs.  79  and  80a,)  the  "added"  component  is  likewise  much  greater 
than  the  true  primary  "  transformer  "  exciting  current.  Thus 
the  total  quadrature  exciting  watts  are  equal  to  the  sum  of  the 
watts  which  would  be  required  for  producing  the  same  mag- 
netic field  by  means  of  two-phase  currents  in  coils  wound  sym- 
metrically on  poles  1,  2,  3  and  4,  and  not  necessarily  equal  to 
twice  the  value  initially  taken  by  the  windings  on  poles  1  and  3. 

SECONDARY  CURRENTS  IN  THE  SINGLE-PHASE  MOTOR. 

It  is  instructive  to  investigate  the  conditions  which  would 
exist  if  the  two  components  of  secondary  current  at  synchronous 
speed  could  be  caused  to  continue  to  flow  unaltered  with  the 
primary  on  open  circuit.  As  noted  above,  the  "  speed  field  " 
current  and  that  component  of  the  secondary  current  which 
tends  to  oppose  the  transformer  field  flow  in  "  electrical  time 
quadrature  "  and  occupy  positions  in  "  electrical  space  quad- 


144  ALTERNATING  CURRENT  MOTORS. 

rature  ";  thus,  if  acting  without  opposition,  they  would  produce 
a  rotating  magnetic  field.  It  is  a  curious  fact,  easily  appreciated 
from  a  study  of  Figs.  79  and  SOa,  that  this  magnetic  field  would 
travel  around  the  air-gap  in  a  direction  opposite  to  the  motion 
of  the  rotor.  Since  the  two  exciting  components  of  the  sec- 
ondary current  in  reality  combine  in  the  rotor  structure  to 
produce  a  resultant  single  current  distributed  throughout  the 
several  conductors,  it  may  be  stated  that  a  band  of  secondary 
exciting  current  revolves  synchronously  in  a  negative  direction. 
If  one  considers  the  time  value  of  the  current  in  a  single  rotor 
conductor,  he  will  discover  that  at  synchronous  speed  this  cur- 
rent is  of  double  frequency.  As  will  be  shown  below,  at 
other  speeds  the  "  secondary  exciting  current  "  has  a  value 
proportional  (approximately)  to  the  speed,  and  it  continues  to 
revolve  synchronously  in  a  negative  direction;  thus  the  fre- 
quency of  this  current  in  an  individual  rotor  conductor  is  equal 
to  the  primary  frequency,  fp,  multiplied  by  one  plus  the  speed, 
5,  with  synchronism  as  unity.  That  is,  fs  =  fp  (1  +  5). 

Consider  now  the  effect  of  operating  the  rotor  at  a  speed 
somewhat  below  synchronism.  Since  there  is  no  opposing 
magnetomotive  force  in  line  with  the  "  speed  field  "  the  "  speed 
field  "  component  of  the  rotor  current  acts  as  though  it  alone  occu- 
pied the  secondary  conductors,  and  its  value  is  in  no  way  affected 
by  the  presence  of  any  other  component  of  secondary  current. 
Thus,  the  e.m.f.  necessary  to  force  the  "  speed  field  "  current 
through  the  secondary  conductors  depends  solely  on  the  value  of 
the  "  speed  field  "  component  of  the  rotor  current.  Since  the 
e.m.f.  generated  in  the  secondary  by  the  motion  of  the  conductors 
through  the  "  transformer  field  "  depends  directly  upon  the 
product  of  this  field  and  the  speed,  it  follows  that  a  definite 
percentage  of  this  speed-generated  e.m.f.  is  consumed  in  the 
"  local  "  secondary  impedance  at  all  speeds,  and  that  the  time 
phase  displacement  between  the  speed-generated  e.m.f.  and  the 
transformer  e.m.f.  of  the  "  speed  field  "  is  constant  at  all  times. 
Thus,  the  "  speed  field  "  at  speed,  5,  bears  to  the  "  transformer 
field  "  a  ratio  equal  to  the  product  of  5  and  a  certain  constant 
which  denotes  the  difference  in  value  and  phase  position  of  the 
"  speed  field  "  and  the  "  transformer  field  "  at  exact  synchronism. 
The  significance  of  this  statement  is  that  the  "  speed  field  " 
component  of  the  secondary  current  has  a  value  proportional 


MAGNETIC  FIELD.  145 

accurately  to  the  product  of  the  speed,  5,  the  "  speed  field  " 
current  at  synchronism  and  the  ratio  of  the  "  transformer  field  " 
at  speed,  5,  to  that  at  synchronous  speed. 

Since  the  transformer  e.m.f.  of  the  "  transformer  "  field  in 
the  rotor  depends  upon  the  strength  of  this  field,  but  is  independ- 
ent of  the  rotor  speed,  while  the  opposing  speed  e.m.f.  of  the 
"  speed  field  "  varies  with  the  product  of  the  speed  and  the 
"  speed  field  "  it  follows  that  the  resultant  e.m.f.  which  tends 
to  produce  "  power  "  current  in  the  secondary  at  speed,  5,  is 
equal  (approximately)  to  the  product  of  the  quantity  (1 — S2) 
and  the  transformer  e.m.f.  (See  Fig.  80b.)  This  component  of 
secondary  current  occupies  at  all  times  a  space  position  mag- 
netically in  line  with  the  "  transformer  field,"  and  it  reacts 
upon  the  primary  just  as  though  it  flowed  through  the  secondary 
of  a  stationary  transformer  into  a  non-inductive  (fictitious) 
load  resistance;  it  is  superposed  in  space,  but  not  in  time,  upon 
that  component  of  the  "  revolving  secondary  exciting  current  " 
which  directly  opposes  the  "transformer  field." 

In  Fig.  80b  let  the  line  0  A  represent  the  value  of  the  trans- 
former e.m.f.,  Et,  of  the  "transformer  field  "  in  the  rotor;  Et 
varies  directly  with  the  "  transformer  field,"  and  hence  decreases 
as  the  primary  current  increases.  Let  the  angle  A  O  B  represent 
the  time  phase  difference  between  Et  and  Es,  the  speed  e.m.f. 
of  the  "  speed  field  "  in  the  rotor;  the  angle  A  0  B  is  constant 
at  all  speeds.  At  synchronous  speed,  Es  has  a  value  0  C  such 
that  the  resultant  of  Et  and  Es  gives  the  electromotive  force, 
Er,  which  produces  that  component  of  rotor  current  which  re- 
acts upon  the  primary.  At  some  lower  speed,  5,  Es  has  a  value 
O  Cv  such  that  O  Ci  =  S2  (0  C),  neglecting  the  relative  decrease 
in  the  value  of.  Et,  and  the  resultant  electromotive  force  which 
produces  current  to  react  upon  the  primary  is  shown  by  A  C \. 
Of  this  latter  e.m.f.  the  component,  C\  Dlt  in  time  quadrature 
with  the  transformer  e.m.f.,  varies  with  S2,  the  square  of  the 
speed;  (that  is,  it  decreases  when  S  decreases),  while  the  com- 
ponent, A  Dv  in  time  phase  with  the  transformer  e.m.f.,  varies 
with  (1 — S2) ;  that  is,  it  increases  with  decrease  of  speed.  To 
the  latter  of  these  components  may  be  attributed  the  secondary 
"  load  "  current,  while  to  the  former  may  be  attributed  that 
component  of  the  "  negatively  revolving  exciting  current " 
which  directly  opposes  the  "  transformer  field." 


146  ALTERNATING  CURRENT  MOTORS. 

When  the  rotor  is  stationary  the  "  load  "  component  of  the 
secondary  current  in  the  individual  conductors  is  of  the  pri- 
mary frequency,  at  nearly  synchronous  speed  it  pulsates  in 
value  in  each  separate  rotor  conductor,  being  unidirectional  in 
certain  conductors  and  alternating  at  double  frequency  in  cer- 
tain other  conductors  situated  90  electrical  space  degrees  from 
the  former. 

It  is  seen,  therefore,  that  there  exist  in  the  rotor  three  com- 
ponents of  secondary  current,  each  of  the  primary  frequency 
with  reference  to  space  representation:  the  "  speed  field  "cur- 
rent, the  current  having  a  value  closely  equal  to  the  product 
of  the  "  speed  field  "  current,  and  the  speed,  but  displaced 
therefrom  both  in  space  and  in  time  by  90  electrical  degrees, 
and  the  load  current.  Each  of  these  varies  in  value  with  the 
"  transformer  field."  The  first  varies  directly  with  the  speed 
5.  The  electromotive  force  which  produces  the  second  varies 
with  S2,  while  the  electromotive  force  which  produces  the  third 
varies  with  (1 — S2).  At  synchronous  speed  the  first  two  com- 
ponents are  equal  in  value,  while  the  third  is  practically  zero. 
At  zero  speed  the  first  two  components  are  zero  and  only  the 
third  flows  in  the  rotor.  Under  all  conditions  the  second  and 
third  components  combine  to  form  the  secondary  current  of 
the  machine  considered  as  a  transformer;  the  second  compo- 
nent acts  as  a  continually  decreasing  (with  decrease  of  speed) 
wattless  current,  while  the  third  acts  in  all  respects  as  though 
it  flowed  through  the  secondary  into  a  non-inductive  load  re- 
sistance. 

GRAPHICAL  REPRESENTATION  OF  SECONDARY  QUANTITIES. 

The  relations  which  exist  between  the  several  components 
of  the  fluxes,  the  currents  and  the  electromotive  forces  in  the 
rotor  at  various  speeds  are  shown  graphically  in  Figs.  81a  and  81b. 
In  Fig.  8 la,  let  the  distance,  A  D,  be  given  an  arbitrary  value 
of  unity,  and  let  the  curve,  A  E  F  D,  be  a  semi-circle.  Then 
if  D  E  is  made  equal  to  the  speed,  5,  B  D  is  equal  to  S2.  Con- 
sider the  condition  when  the  speed,  S,  has  the  value  represented 
by  F  D\  the  ratio  of  the  "  speed  field  "  to  the  "  transformer 
field  "  is  shown  directly  by  the  line,  F  D\  this  line  also  shows 
the  ratio  of  the  true  "  speed  field  "  current  to  the  true 
"  transformer  field  "  current,  and  likewise  the  ratio  of  the 


MAGNETIC  FIELD. 


147 


component  of  the  secondary  current  which  directly  opposes 
the  "  transformer  field  "  to  the  true  "  speed  field  "  current. 
Thus,  if  at  the  speed  shown  by  F  D,  A  D  is  assumed  equal 
to  the  "  transformer  field  "  current,  D  F  is  equal  to  the  true 
"  speed  field  "  current  and  C  D  is  equal  to  the  "opposing" 
component  of  the  secondary  current.  Furthermore  if  at 
the  speed,  F  D,  A  D  be  made  equal  to  the  e.m.f.  which 
would  be  produced  in  the  secondary  by  the  "  transformer 
field  "  with  the  rotor  stationary,  C  D  is  the  actual  speed  e.m.f. 
due  to  the  motion  through  the  "  speed  field,"  and  A  C  is  the 
e.m.f.  which  causes  "  load  "  current  to  flow  through  the  sec- 
ondary impedance. 


Fio.  8lA. — Numerical  value  of       FIG.  81s. — Time  and  Space  Values 
Currents  and  Electromotive  Forces.  of  Fluxes  and  Currents. 


It  is  to  be  noted  especially  that  the  diagram  of  Fig.  81  a  gives 
only  the  relative  numerical  values  of  the  various  components 
and  does  not  indicate  their  time-phase  or  electrical  space  po- 
sitions. The  electrical  space  and  time  values  of  the  electro- 
motive forces  are  shown  in  Fig.  80b,  while  the  equivalent  values 
for  the  fluxes  and  currents  are  represented  in  Fig.  81  b.  In  this 
diagram  G  H  is  equal  to  A  D  of  Fig.  81a,  while  the  curve,  G  L HI, 
is  a  circle;  J  K  is  made  equal  to  F  D  and  the  curve,  G  K  H  J, 
is  an  ellipse;  M  N  is  equal  to  C  D  and  curve,  M  K  N  J,  is  an 
ellipse.  The  electrical  space  value  of  the  flux  at  synchronous 
speed  is  shown  by  circle,  G  L  H  I  while  at  speed,  D  F,  it  has 
the  value  indicated  by  ellipse,  G  K  H  J.  If  the  line,  G  H, 


'148  ALTERNATING  CURRENT  MOTORS 

shows  the  value  and  phase  position  of  the  true  "  transformer 
field"  current,  the  line,  /  K,  simultaneously  shows  the  value 
and  phase  position  of  the  true  "  speed-field  "  current;  these  cur- 
rents are  in  separate  electrical  structures,  and  they  do  not  com- 
bine directly,  but  their  magnetomotive  forces  combine  to  pro- 
duce the  elliptical  revolving  magnetic  field.  The  value  and 
phase  position  of  the  "  opposing  "  component  of  the  secondary 
current  is  shown  by  the  line,  N  M;  this  current  is  in  the  same 
electrical  structure  with  the  current,  /  K,  and  the  two  combine 
to  produce  the  "  negatively  revolving  secondary  exciting  cur- 
rent," shown  by  curve,  K  M  J  N,  which  is  elliptical  as  to  space 
representation. 

It  is  interesting  to  observe  that  the  actual  "  speed  field  "  cur- 
rent in  the  secondary  varies  directly  with  the  speed,  but  that 
the  component  of  the  secondary  current  which  reacts  directly 
upon  the  transformer  field  varies  with  the  square  of  the  speed, 
or,  more  correctly,  with  the  square  of  the  transformer  field. 
It  will  be  noted  that  on  account  of  this  fact  the  total  "  quad- 
rature exciting  watts "  of  the  single-phase  induction  motor 
vary  directly  with  the  square  of  the  "  transformer  field  "  plus 
the  square  of  the  "  speed  field."  Thus  the  true  exciting  watts 
of  the  machine  at  any  speed  are  directly  proportional  to  the 
sum  of  the  squares  of  the  densities  of  the  fluxes  traversing  the 
several  magnetic  paths,  as  was  mentioned  in  the  last  chapter. 


CHAPTER  XI. 
SYNCHRONOUS  MOTORS  AND  CONVERTERS. 

SYNCHRONOUS  COMMUTATING  MACHINES. 

The  term  "  synchronous  commutating  machines  "  refers  to 
all  motors  or  generators  which  receive  or  deliver  both  alter- 
nating and  direct  current.  The  machines  discussed  below  are 
rotary  converters  and  double-current  generators,  and  compari- 
sons are  made  with  the  capacities  of  alternating-current  gen- 
erators or  motors  of  different  number  of  phases. 

For  simplicity  in  treatment,  the  rotary  converters  are  as- 
sumed to  deliver  at  the  direct-current  commutator  all  of  the 
power  received  at  the  alternating  end;  that  is,  the  output  is 
'assumed  equal  to  the  input  in  determining  the  relative  currents 
'on  each  side,  though,  as  will  be  seen  later,  the  armature  copper 
loss  is  properly  accounted  for.  The  double-current  generators 
:are  assumed  to  deliver  equal  amounts  of  power  at  the  commu- 
tator and  at  the  collector  rings.  The  assumption  is  further 
made  that  the  alternating-current  wave  in  each  case  follows  a 
true  sine  curve  of  time- value. 

In  a  rotary  converter  the  mean  flow  of  alternating  current  is 
in  a  direction  opposed  to  the  flow  of  the  direct  current,  but  the 
absolute  value  of  the  alternating  current  varies  from  time  to 
time  and  the  direct  current  reverses  direction  of  flow  through 
the  individual  coils  as  each  passes  under  one  of  the  brushes, 
so  that  the  resultant  current  in  the  coils  varies  both  in  value 
and  direction  of  flow  from  instant  to  instant  and,  in  general, 
it  has  not  the  same  heating  effect  in  different  armature  coils. 

When  the  alternating  current  has  unity  power  factor,  the 
maximum  value  of  the  current,  evidently,  occurs  when  the 
group  of  coils  of  the  phase  under  consideration  are  developing 
their  maximum  e.m.f.  The  mechanical  position  of  the  coils  at 
this  instant  of  maximum  e.m.f.  is  that  in  which  the  center  of 
the  group  of  coils  is  passing  at  right  angles  to  the  lines  of  force 
from  a  field  pole — with  a  non-distorted  field  this  position 

149 


150 


ALTERNATING  CURRENT  MOTORS. 


would  be  opposite  the  center  of  the  field  pole.  When  the  coils 
are  passing  parallel  to  the  lines  of  force  the  e.m.f.  is  of  course 
zero.  At  intermediate  positions,  the  value  of  the  e.m.f.  may 
be  represented  by  Em  cos  0,  where  Em  represents  the  maximum 
e.m.f.  and  0  the  angle  between  the  instantaneous  position  of 
the  coils  and  a  line  from  the  pole  center  to  the  center  of  the 
armature  shaft. 

The  absolute  value  of  the  maximum  e.m.f.  depends  upon  the 
number  of  coils  in  the  group  considered.  While  the  effective 
value  of  the  e.m.f.  developed  in  each  coil  is  the  same  as  that 
in  the  others,  and  adjacent  coils  are  connected  in  series,  the 
effective  value  of  the  e.m.f.  of  a  group  of  coils  is  not  proportional 


FIGS.  82A  and  82e. — Phase  Relations  of  Voltages. 

directly  to  the  number  of  coils  composing  a  group,  since  the 
e.m.f.  of  one  coil  is  not  directly  in  phase  with  that  of  the  adja- 
cent coils;  that  is,  the  e.m.f.  of  each  coil  reaches  its  maximum 
value  at  a  different  instant  from  that  corresponding  to  the 
maximum  e.m.f.  of  each  of  the  other  coils. 

If  time  be  represented  as  angular  degrees  passed  over  by  the 
armature  of  a  bipolar  machine,  and  the  value  of  the  e.m.f.  of 
each  individual  coil  be  denoted  by  a  line  of  any  chosen  length, 
and  the  line  for  each  coil  be  placed  in  the  angular-time  position 
which  the  armature  would  occupy  when  that  coil  has  its  max- 
imum e.m.f.,  a  diagram  similar  to  that  represented  by  Fig.  82a 
will  be  produced.  Here  01  represents  the  effective  value  and 
time  position  of  the  e.m.f.  in  coil  No.  1,  and  02  represents 


MOTORS  AND  CONVERTERS.  151 

corresponding  quantities  for  coil  No.  2,  etc.  As  stated  above; 
these  coils  are  connected  in  series,  so  that  the  actual  effective 
value  of  the  e.m.f.  of  the  coils  as  interconnected  may  be  repre- 
sented as  in  Fig.  82b.  It  will  be  observed  that  as  the  number 
of  coils  is  increased  the  figure  approaches  a  circle  and  that  in 
any  case  the  extremities  of  the  sides  lie  on  a  circle. 

By  the  use  of  the  figure  below  or  equivalent  circle  it  is  a 
simple  matter  to  determine  the  effective  value  of  the  e.m.f.  of 
a  group  of*  coils  on  an  armature.  This  e.m.f.  is  seen  to  be 
represented  in  value  by  the  chord  of  the  arc  subtended  by  the 
group  of  coils.  If  the  total  number  of  coils  on  the  armature 
be  divided  into  P  equal  parts,  then  the  angle  covered  by  each 
part  is  360°  -r-P;  and,  since  the  chord  is  equal  to  twice  the  sine 
of  half  the  angle,  the  e.m.f.  of  each  group  is 

0£     .    180°  180° 

EP  =  Z-^sw—p-   =  Esm—— 

where  E  is  the  value  of  the  e.m.f.  measured  across  a  diameter. 
Now,  E  is  the  effective  value  of  the  e.m.f.  at  the  diameter, 
while  for  a  rotary  converter  the  direct-current  commutated  e.m.f. 
is  equal  to  the  maximum  value  of  this  e.m.f.,  or  is  \/~2E  =  Em. 
Therefore,  the  effective  maximum  value  of  the  e.m.f.  of 

a  group  of    coils   which   cover  —  part  of  the  armature  is  equal 


to 


Em    .    180°       „ 
„„  __  _  £,. 


When  the  alternating  current  is  in  phase  with  the  e.m.f., 
the  product  of  the  current  flowing  in  the  coils  selected,  by  the 
e.m.f.  across  the  group  gives  the  power  in  watts  in  that  section 
of  the  armature  and  when  the  armature  is  symmetrically  loaded, 
the  total  power  is 

180° 

C4/M 

V2 


W  =  P  IP  ^  sin 


SYNCHRONOUS  MOTORS  AND  GENERATORS. 

For  the  purpose  of  subsequently  comparing  the  capacities  of 
alternating-current  machines  of  various  types  and  phases,  it  is 
convenient  at  this  point  to  ascertain,  by  means  of  the  above 
formula,  the  relative  capacities  of  a  closed-coil  armature  used 


152  ALTERNATING  CURRENT  MOTORS. 

in  a  direct-current  generator  and  the  same  armature  used  in 
alternating-current  generators  of  different  number  of  phases. 
Consider  the  armature  to  revolve  in  a  field  of  constant  intensity 
at  a  constant  speed.  There  will  be  generated  the  same  e.m.f. 
per  conductor  irrespective  of  the  connections  of  the  external 
circuits.  Assume  that  the  capacity  is  in  each  case  wholly  de- 
termined by  the  heating  of  the  armature  conductors  and,  as  a 
method  of  direct  comparison,  assume  that  the  external  load  is 
so  adjusted  in  each  case  that  there  flows  the  same  current  through 
each  conductor  on  the  armature  whether  used  in  a  direct  or  an 
alternating-current  generator  and  independent  of  the  number 
of  phases.  Obviously,  the  loss  from  heating  of  the  armature 
conductors  will  always  remain  the  same,  while  the  capacity 
will  vary  as  the  external  load. 

TABLE  I. 
Capacities  of  Alternator  Compared  to  Direct-Current  Generator  as  100. 


Number 
of 
Phases. 

Volts 
Between 
Leads 

Amperes 
Per 
Phase. 

Total 
Output. 

Amperes 
Per 
Lead. 

(Rings.) 

100    .     180 

—-r-  sm  — 
x/2          p 

PEp  IP 

2  Wt 
70.71  P 

P 

EP 

IP 

Wt 

IL 

2 

70.71 

5 

707.1 

10.00 

3 

61.24 

5 

918.6 

8.66 

4 

50.00 

5 

1000.0 

7.07 

6 

35.35 

5 

1060.6 

5.00 

Infinite 

0.4- 

5 

1110.7 

0.4- 

For  simplicity  in  comparison  assume  that  there  flows  always 
5  amperes  in  each  armature  conductor,  and  also  that  the  e.m.f. 
measured  between  the  direct-current  brushes  is  100.  The 
capacity  as  a  direct-current  generator  is  evidently  1000  watts, 
while  the  outputs  as  alternating-current  generators  of  various 
numbers  of  phases  will  be  as  in  Table  I.  (See  also  Fig.  83a.) 

When  P  =  infinity,  EP  =  0,  but  P  Ep=  100X\/4  X7r'  as 
will  be  shown  later. 

It  is  interesting  to  note  that  the  capacity  of  an  alternating- 
current  generator  can  be  represented  as  the  perimeter  of  a  polygon 
having  sides  equal  in  number  to  the  number  of  phases,  of  which 
polygon  the  circumscribing  circle  represents  the  capacity  for 
infinite  phases,  double  the  diameter  of  this  circle  representing 


MOTORS  AND  CONVERTERS. 


153 


the  capacity  of  the  so-called  single-phase  generator,  while  the 
capacity  of  the  machine  as  a  direct-current  generator  is  repre- 
sented in  value  by  the  perimeter  of  a  square  inscribed  within 
.the  circle.  These  facts  will  be  brought  out  by  an  inspection 
of  Fig.  83a. 

It  is  to  be  observed  that  the  output  given  above  is  the  volt- 
ampere  capacity  of  each  machine.  With  an  alternating-current 
_  generator,  the  power  delivered  will,  of  course,  vary  with  the 
power  factor.  At  any  power  factor  less  than  unity  the  ratio  of 
the  alternating  to  the  direct-current  capacities  would  vary 


2-Phase-(se-eal-led-Si-Eg-le-Phase-70r7-l- 


FIG.   83A. — Relative  Currents  for  Same  Heat  Loss 
closed  Coil  Generator  Armature. 


En- 


directly  therewith,  but  the  ratio  of  the  alternating-current 
capacities  for  different  numbers  of  phase's  would  remain  the 
same  independent  of  the  power  factor. 

In  the  equations  given,  P  corresponds  to  the  number  of  col- 
lector rings.  Thus,  a  closed-coil  single-phase  generator,  so- 
called,  is  considered  a  two-phase  generator,  the  phases  being 
180°  apart.  A  so-called  two-phase  generator  having  a  closed- 
coil  armature  with  four  collector  rings  is,  in  fact,  a  four-phase 
generator  with  phases  90°  apart,  though  its  capacity  is  neither 
increased  nor  decreased  by  loading  as  two  separate  two-phase 
(so-called  single-phase)  generators. 


154  ALTERNATING  CURRENT  MOTORS. 

SYNCHRONOUS  CONVERTERS,  UNITY  POWER-FACTOR. 

The  problem  of  determining  the  relative  capacities  of  rotary 
converters  and  other  synchronous  commutating  machines  can 
be  attacked  by  use  of  the  same  fundamental  equation  developed 
above  as  applied  to  the  alternating-current  generators,  though 
the  method  of  application  must  be  slightly  modified  to  suit 
the  various  types  of  machines.  Perhaps  the  simplest  method 
of  ascertaining  the  effect  of  the  presence  of  both  the  direct  and 
the  alternating  current  upon  the  relative  copper  loss  of  the 
armature  is  to  compare  the  losses  of  different  machines  for 

TABLE  II. 
Volts  and  Amperes  for  Same  Power  with  Different  Numbers  of  Phases. 


Number 
of 
Phases. 

Volts 
Between 
Leads. 

Amp.  per 
Phase;  Ef- 
fective. 

Amp.  per 
Phase; 
Max. 

Amp.  per 
Lead;  Ef- 
fective. 

100    .     180 

W 

V2IP 

VX2W 

P.  EP 

P.  E 

P 

=  EP 

=  Ip 

=  lM 

-It 

2 

70.71 

7.071 

10.000 

14.142 

(single-  phase) 

3 

61.24 

5.443 

7.698 

9.428 

4 

50.00 

5.000 

7.071 

7.071 

(tsvo-  phase) 

6 

35.35 

4.714 

6.665 

4.714 

Infinite        ; 

0.4- 

4.501 

6.365 

0.4- 

D.  C. 
App.  2 

E  =  100 

I-., 

4  =  5.0- 

1  =  10 

assumed  equal  outputs,  and  then  to  determine  the  relative 
outputs  for  the  same  loss. 

The  formula  referred  to  above  enables  one  to  determine  at 
once  the  effective  value  of  current  which  is  necessary  to  give 
a  certain  amount  of  power  when  P,  the  number  of  phase,  and 
Em,  the  direct  e.m.f.  are  known.  Table  II  gives  the  value  of 
current  for  various  number  of  phases  for  an  assumed  power  of 
1000  watts  and  direct  e.m.f.  of  100  volts. 

It  is  to  be  noted  that  as  the  number  of  phases  increases  the 
current  per  group  of  coils  decreases,  but  that,  even  with  an 
infinite  number  of  phases,  the  current  has  yet  a  finite  value. 


MOTORS  AND  CONVERTERS. 


155 


An  inspection  of  Fig.  83a  will  show  that  the  total  e.m.f.  of  the 
infinity  groups  of  infinity  phases  is  represented  by  the  circum- 
ference of  the  circumscribing  circle.  The  value  of  current  to 
produce  the  assumed  power  is  found  by  dividing  the  1000 
watts  by  this  total  e.m.f. 

The  maximum  value  of  current  for  sine  waves  is  \/2"  times 
the  effective  value  and,  when  the  power  factor  is  unity,  this 
maximum  current  flows  when  the  coils  are  developing  their 
maximum  e.m.f.  With  an  armature  in  a  bipolar  field,  as  shown 
in  Fig.  83b,  the  maximum  value  of  e.m.f.  in  a  group  of  coils  occurs 


10  Amperes  _ 


FIG.  83s. — Current  in  Armature  Conductors  of  Four-phase 
Synchronous  Converter. 

at  that  position  of  the  revolution  of  the  armature  where  the 
line  joining  the  extremities  of  the  group  is  in  a  vertical  plane, 
and  the  e.m.f.  in  other  positions  varies  as  the  cosine  of  the  angle 
of  deviation  from  the  vertical  position. 

Having  determined  the  value  of  the  maximum  alternating 
current  and  the  position  of  the  group  of  coils  when  this  max- 
imum flows  it  now  remains  to  investigate  the  effect  of  the 
presence  of  the  direct  current  in  the  armature  coils. 

For  purpose  of  combined  generality  of  treatment  and  sim- 
plicity of  discussion,  the  so-called  single- phase  rotary  will  be 
omitted  for  the  present  and  there  will  be  discussed  first  the  so- 


156  ALTERNATING  CURRENT  MOTORS. 

called  two-phase  machine  which  is  in  reality  a  four-phase, 
rotary  converter.  Since  there  are  four  phases,  the  group  of 
coils  for  each  phase  covers  90  degrees.  Assume  that  there  are 
72  coils  on  the  armature.  There  will  then  be  18  coils  per  phase, 
and  each  coil  covers  5°.  (The  treatment  here  given  is  general 
and  results  will  be  in  no  way  affected  if  the  5°  contain  any 
number  of  coils,  or  in  fact,  less  than  one  coil.) 

Consider  the  instant  when  the  group  of  coils  is  in  the  position 
at  which  the  maximum  e.m.f.  is  generated,  as  indicated  in  Fig. 
83b.  The  alternating  current  is  equal  to  \/2  I  —  7.071,  while 
the  direct  current  is  500-:- 100  =  5,  so  that  the  actual  current 
flowing  through  the  coils  is  7.071  —  5,  causing  a  relative  loss  of 
(2.07)2X18  =  77,  where  the  resistance  of  each  coil  is  taken  as 
unity. 

As  the  armature  moves  forward  5°  the  alternating  current  drops 
to  7.071  cos  5°  =  7.05,  while  the  direct  current  remains  at  5, 
causing  a  relative  loss  of  (2.05)2X18  =  76.  In  this  manner 
the  relative  loss  for  each  position  of  the  armature  may  be  de- 
termined up  to  that  number  of  degrees  rotation  which  brings 
the  beginning  of  the  group  of  coils  under  the  -f  brush.  When 
the  armature  has  rotated  50°  one  coil  of  the  group  considered 
will  be  on  the  right  of  the  brush,  and,  though  this  coil  has  the 
same  value  of  alternating  current  in  it  as  has  each  of  the  others 
of  the  group,  the  direct  current  through  it  is  reversed  land  the 
resultant  current  is,  therefore,'  greater  than  in  the  other  coils 
or  is  equal  to  4.55  +  5  =  9.55,  causing  a  relative  loss  of  (9.55)2 
Xl  =  91.2.  The  other  coils  have  at  this  instant  a  resultant 
current  of  4.55  —  5  and  a  relative  loss  of  (—  .45)2X17  =  3.4, 
hence  the  total  relative  loss  for  the  group  is  91.2  +  3.4  =  94.6. 

As  the  armature  continues  to  rotate,  more  coils  pass  into  the 
right-hand  section  and  less  remain  in  the  left-hand  section,  till,, 
when  the  armature  has  rotated  90°  from  its  initial  position,  the 
coils  are  equally  divided  between  the  two  sections — one-half 
on  each  side  of  the  brush.  At  this  instant  the  alternating  cur- 
rent will  have  decreased  to  zero  and  the  total  relative  loss  will 
be  450,  which  is  the  loss  due  to  the  direct  current  alone. 

Continuing  this  investigation  till  the  armature  has  rotated 
180°,  it  will  be  plain  that  the  conditions  obtained  at  the  begin- 
ning are  being  repeated,  so  that  a  mean  of  the  total  relative 
losses  throughout  the  180°  is  the  same  as  occurs  continuously, 


MOTORS  AND  CONVERTERS. 


157 


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Ul   SSOJ   3AI 


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oo! 


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JO    9n[13A 


0 


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158 


ALTERNATING  CURRENT  MOTORS. 


throughout  the  operation  of  the  rotary.  As  shown  by  Table  III, 
this  mean  relative  loss  is  170,  for  the  conditions  herein  assumed. 
The  relative  loss  for  the  group  of  coils  if  the  machine  were 
operating  as  a  direct-current  generator  would  be  450,  as  found 
above.  The  relation  between  these,  1  to  2.647,  indicates  the 
relative  loss  of  the  four-phase  rotary  converter  compared  with 
the  corresponding  direct-current  generator,  at  the  same  output, 
as  unity.  Since  the  loss  in  any  circuit  varies  as  the  square  of 
the  current,  the  relative  currents  to  give  the  same  loss  should 
vary  as  the  square  root  of  the  above  ratio,  or  the  relative  ca- 

Tlme-  Degrees  from  Point  of  Maximum  E.M.F.  and  or  Maximum  Current 

10   20  30   40   50  60   70  80   80  100  110  120  180  140  150  160  170  180  100  200  210  220  2*> 


£„ 


rr1=fm±rjd 


lA 


£- 


Hg. 


FIG.  84.  —  Distribution  of  Loss  in  Armature  of  Four-phase 
Synchronous  Converter,  the  Angle  of  Lag  Being  Zero. 


pacities  of  the   machine   as   a   rotary   and   as   a   direct-current 
generator  will  be  \/2.647  =  1-627. 

An  inspection  of  columns  4  and  5  of  Table  III  or  of  equivalent 
curves  of  Fig.  84  reveals  the  manner  in  which  the  instantaneous 
value  of  current  in  the  coils  varies.  It  will  be  seen  that,  though 
the  mean  effective  value  of  current  for  the  group  of  coils  is  less 
as  a  rotary  than  as  a  direct-current  generator,  there  are  certain 
coils  which  at  certain  times  carry  more  current  than  others 
and  that  one  coil  will  carry  a  maximum  of  twice  the  current 
when  operating  as  a  four-phase  rotary  as  a  direct-current 
generator. 


MOTORS  AND  CONVERTERS.  159 

DISTRIBUTION  OF  HEAT  Loss  IN  ARMATURE  COILS. 

As  a  result  of  the  variation  in  strength  of  the  alternating 
current  at  the  instant  when  each  separate  armature  coil  of  a 
rotary  converter  or  a  double-current  generator  passes  under  a 
commutating  brush,  at  which  time  the  direct  current  within 
the  coil  is  reversed,  the  maximum  value  of  current  to  which  a 
coil  is  subjected  varies  with  the  individual  coils  according  to 
the  location  of  each  within  the  group  constituting  the  windings 
of  one  phase — the  windings  between  two  adjacent  collector-ring 
taps  in  a  bipolar  armature.  The  two  end-coils,  that  is,  the  coils 
which  connect  the  alternating-current  leads  to  the  adjacent 
groups  on  either  side,  carry  greater  values  of  current  than  the 
contiguous  coils  within  the  group,  but  these  two  coils  do  not  have 

TABLE  IV. 


Type  of 
Machine. 

Rotary 
Converter. 

Double-Current 
Generator. 

Number 
of 
Phases. 

100% 
Power 
Factor. 

90.63% 
Power 
Factor. 

100% 
Power 
Factor. 

90.63% 
Power 
Factor. 

2 

3.00 

3.21 

1.50 

1.60 

3 

2.33 

2.70 

1.27 

1.35 

4 

2.00 

2.47 

1.21 

1.28 

6 

1.67 

2.21 

1.17 

1.24 

Infinite 

1  .00 

1.60 

1.14 

1  .20 

equal  current  values  when  the  power  factor  of  the  alternating 
current  is  less  than  unity. 

A  knowledge  of  the  relative  increase  in  instantaneous  value 
of  maximum  current  is  important  and  a  study  of  its  effect  and 
location  as  to  coils  is  instructive  as  indicating  the  existence  of 
local  heating  within  the  armature  windings.  The  value  of  this 
maximum  current  which  flows  within  a  single  coil  depends  upon 
the  number  of  phases  and  the  power  factor  of  the  current. 
Table  IV  gives  the  relative  values  of  this  maximum  current  for 
different  machines  considering  as  unity  the  current  in  a  direct- 
current  generator  nt  the  same  load. 

Although  the  alternating  current  follows  a  sine  wave,  the 
current  in  individual  coils  does  not  follow  a  sine  curve  of  time- 
value,  and  compared  to  its  effective  heating  value,  the  max- 
imum value  is  much  greater  than  that  obtained  with  a  true 


160 


ALTERNATING  CURRENT  MOTORS 


sine  curve.  The  maximum  current  flows  for  only  a  small  frac- 
tion of  the  total  time  and  is  confined  to  a  relatively  small  por- 
tion of  the  armature,  so  that  the  excess  heating  effect  cannot  at 
once  be  judged  from  Table  IV,  but  must  be  determined  by 
calculation  similar  to  those  recorded  in  columns  11  and  12  of 
Table  III. 

Table  V  indicates  the  relative  values  of  the  maximum  and 
the  minimum  losses  in  individual  coils  on  the  armature  of  syn- 
chronous commutating  machines  of  various  phases  at  power 
factors  of  100  per  cent,  and  of  90.63  per  cent.,  compared  to  the 
mean  armature  loss  per  coil  under  the  same  condition  of  service. 
The  results  here  recorded,  therefore,  indicate  the  relative  lack 
of  uniformity  of  distribution  of  heat  loss  in  the  armature  wind- 
ings. 

TABLE  v. 

Maximum  and  Minimum  Losses  in  Individual  Coils. 


Type  of 
Machine. 

Rotary  Converter. 

Double-Current  Generator. 

Number 
of 
Phases. 

100% 
Power  Factor. 
Max.         Min. 

90.63% 
Power  Factor. 
Max.           Min. 

100% 
Power  Factor. 
Max.           Min. 

90.63% 
Power  Factor. 
Max.           Min. 

2 

2.270 

.331 

2.462 

.334 

1.201 

.650 

1.462 

.443 

3 

2.161 

.405 

2.748 

.343 

1.084 

.828 

1.132 

.647 

4 

1.926 

.531 

2.600 

.391 

1.048 

.903 

1.094 

.753 

6 

1.590 

.725 

2.217 

.461 

1.018 

.955 

1.065 

.852 

Infinite 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

SYNCHRONOUS  CONVERTERS,  FRACTIONAL  POWER  FACTOR.- 

When  a  rotary  converter  is  operated  at  a  power  factor  less 
than  unity  two  effects  are  observed:  There  is  required  a  propor- 
tionately larger  current  to  produce  a  given  power,  and  the 
maximum  current  does  not  flow  in  a  given  group  of  coils  when 
the  coils  are  generating  their  maximum  e.m.f.  Though  the 
relative  loss  for  a  given  machine  does  not  vary  regularly  with 
decrease  in  power  factor,  it  is  sufficient  for  present  purposes  to 
determine  the  effect  of  operating  the  machines  at  a  single  fairly 
low  value  of  power  factor. 

Assume  that  the  current  lags  25°  behind  the  rotary  e.m.f. 
The  power  factor  is,  therefore,  cos  25°  =  .9063  and  the  max- 
imum value  of  current  instead  of  being  7.071,  as  before  for  the 


MOTORS  AND  CONVERTERS. 


161 


four-phase  rotary  converter  is  now  7. 071  -^-.9063  =  7.81  am- 
peres, and  this  maximum  occurs  when  the  armature  has  moved 
forward  25°  from  the  position  giving  maximum  e.m.f.,  or,  what 
is  the  same  thing,  the  armature  must  now  rotate  forward  only 
20°  before  the  group  of  coils  begins  to  pass  under  the  brush 
instead  of  45°,  as  when  the  current  and  e.m.f.  are  in  phase. 

Bearing  these  facts  in  mind  and  making  proper  substitution 
in  Table  III,  there  is  obtained  by  a  method  similar  to  the  one 
used  previously  a  mean  relative  copper  heat  loss  in  the  group 
of  coils  of  266  (Table  VI)  which,  compared  to  the  loss  of  450 

Time-  Degrees  from  Point  Maximum  E.M.F. 

0   10   20   30   40   60   00  70   80   90  100  110  120  180  140  150  160  170  180  190  200  210  220  8W 

Degrees  from  Point  of  Maximum  Current 

0   10   20   30  40   50   60   70   80   90  100  110  120  180  140  160  160  170  180  190  200 


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FIG.  85. — Distribution  of  Loss  in  Armature  of  Four-phase 
Synchronous  Converter  at  25°  Lag. 


for  the  direct-current  generator,  indicates  a  relative  output  of 

450 

-  1.300  as  compared  with  that  of  the  direct-current  gen- 


erator. 

DOUBLE  CURRENT  MACHINES. 

The  method  of  determining  the  output  from  the  double-cur- 
rent generator  is  quite  similar  to  that  used  above.  In  this 
case,  however,  the  direction  of  flow  of  the  alternating  current 
at  the  time  of  maximum  value  is  the  same  as  that  of  the  direct 


162 


ALTERNATING  CURRENT  MOTORS. 


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MOTORS  AND  CONVERTERS. 


163 


current,  so  that  the  sum,  and  not  the  difference,  must  be  taken 
in  determining  the  resultant  current.  Columns  13  and  14  of 
Table  III  will  show  the  manner  in  which  the  instantaneous  value 
of  resultant  current  varies.  The  relative  loss  of  1631.6  is  for 
equal  direct  and  alternating-current  outputs  so  that  the  rela- 
tive total  output  for  the  same  loss  as  in  a  direct-current  gen- 
erator will  be 


Table  VI  records  the  calculation  for  determining  the  effect 
of  a  lag  of  25°  in  the  alternating  current  for  the  double-current 
generator,  and,  as  the  method  is  quite  the  same  as  used  for 
the  rotary  converter,  it  need  not  be  further  discussed. 

TABLE  VII. 
Relative  Capacities  of  Alternating-Current   Machines;   Direct-Current  Capacity  =  1. 


Type  of 
Machine. 

Rotary  Converter. 

Double-Current  Gen. 

Alt.  -Cur.  Generator. 

Number 
of 
Phases. 

100% 
Power 
iv  actor. 

90.63% 

Power 
Factor. 

100% 
Power 
Factor. 

90.63% 
Power 
Factor. 

100% 
Power 
Factor. 

90.63% 
Power 
Factor. 

2 

.848 

.731 

.951 

.890 

.7071 

.6418 

3 

1.338 

1.103 

1.023 

.992 

.9186 

.8325 

4 

1.627 

1.300 

1.050 

1.020 

1.0000 

.9063 

6 

1.937 

1.482 

1.066 

1.038 

1  .0610 

.9612 

Infinite 

2  291 

1   648 

1.082 

1.052 

1  .1105 

1.0066 

RELATIVE  CAPACITIES  OF  SYNCHRONOUS  MACHINES  OF  VARIOUS 

PHASES. 

The  relative  capacities  of  alternating-current  machines  com- 
pared with  a  direct-current  generator  as  unity  are  given  in  Table 
VII.  The  results  recorded  in  Table  VII  are  plotted  in  the  form 
of  curves  in  Fig.  86,  so  as  to  show  to  the  eye  the  effect  of  varying 
the  number  of  phases  of  a  given  machine.  It  will  be  seen  at  a 
glance  that  increasing  the  number  of  phases  in  each  case  in- 
creases the  capacity  of  the  machine,  but  that  the  relative  in- 
crease for  alternating-current  and  double-current  generators  is 
small  compared  to  the  increase  for  rotary  converters.  A 
change  from  three  to  six  phases  with  an  alternating-current 
generator  results  in  an  increased  capacity  of  15.5  per  cent., 


164 


ALTERNATING  CURRENT  MOTORS. 


while  an  equivalent  change  with  a  rotary  converter  produces 
from  35  per  cent,  to  45  per  cent,  greater  output,  depending 
•upon  the  power  factor  of  operation.  These  latter  figures  may 
be  increased  or  decreased  if  the  current  wave  departs  materially 
from  the  assumed  sine  curve  of  time-value,  though  the  figures 
as  given  represent  results  obtained  in  practice. 


00  %  Power 


2.34  6  8 

JJumber-of  Phases.       (Collector  Kings) 


Infinite  Phases 


FIG.  86. — Capacities  of  Alternating  Current  Machines  Com- 
pared   to    Direct-current    Generator. 

The  increase  in  capacity  of  a  rotary  converter  resulting  from 
a  change  from  the  so-called  single-phase  to  three-phase  is 
from  51  per  cent,  to  57  per  cent.  Therefore,  changing  a  given 
rotary  from  three-phase  to  six  results  in  from  70  per  cent,  to 
80  per  cent,  as  great  increase  in  capacity  as  changing  from 
single  to  three-phase.  Reference  to  Table  V  will  show  that 


MOTORS  AND  CONVERTERS.  165 

with  a  three-phase  rotary  converter  one  section  of  the  armature 
windings  is  subjected  to  from  5.3  to  8  times  as  great  current 
heating  effect  as  certain  others,  while,  with  a  six-phase  machine, 
the  corresponding  results  are  2.2  and  4.8.  This  means  that 
the  heat  loss  is  much  better  distributed  in  a  six-phase  con- 
verter armature  than  in  a  three-phase  one. 

Since  the  transformers  necessary  to  convert  the  three-phase 
current  from  the  high  potential  transmission  circuits  to  six-phase 
for  rotaries  are  in  no  way  more  expensive  or  complicated  than 
those  for  three-phase  rotaries,  economy  in  cost  of  equipment 
and  efficiency  of  operation  dictates  the  use  of  six-phase  ma- 
chines, and  where  the  size  of  the  rotaries  operated  justified  the 
additional  connecting  circuits  between  the  transformers  and  the 
machines,  six-phase  rotary  converters  should  be  used. 

CHARACTERISTIC  PERFORMANCE  OF  SYNCHRONOUS  CONVERTERS. 

In  external  appearance  a  polyphase  rotary  converter  resem- 
bles a  direct-current  generator  with  a  conspicuously  large  com- 
mutator and  an  auxiliary  set  of  collector  rings.  Its  design  in 
certain  respects  is  a  compromise  between  alternating-current 
and  direct-current  practice.  This  is  most  noticeable  with 
reference  to  the  speed  and  number  of  poles ;  that  is,  the  fre- 
quency. A  careful  review  of  constructive  data  for  modern 
direct-current  railway  generators  reveals  the  fact  that  the 
frequency  of  such  machines  is  between  8  and  12  cycles  while 
the  frequency  of  the  older  belted  type  was  about  20  cycles. 
Alternating-current  generators,  on  the  contrary,  when  not  lim- 
ited in  frequency,  are  seldom  built  for  less  than  60  cycles. 

Since  the  rotary  converter  is  in  fact  a  synchronous  motor,  it 
must  run  at  a  speed  determined  by  the  alternations  of  the  sup- 
ply and  the  number  of  its  poles.  A  limit  to  the  possible  in- 
crease in  speed  of  the  converter  is  set  by  the  peripheral  speed 
of  the  commutator.  Experience  has  demonstrated  that  3000 
ft.  per  minute  is  as  high  as  the  commutator  should  run  to  give 
reliable  service.  The  peripheral  speed  of  a  rotary  converter  is 
equal  to  the  product  of  the  number  of  alternations  by  the  dis- 
tance between  two  adjacent  neutral  points.  For  a  given  direct- 
current  e.m.f.,  a  limiting  potential  difference  of  from  8  to  10 
volts  between  segments,  and  the  minimum  size  of  bars,  allow- 
ing for  insulation,  it  is  evident  that,  with  a  limiting  peripheral 


166  ALTERNATING  CURRENT  MOTORS. 

speed,  there  is  soon  reached  a  limit  to  the  number  of  alterna- 
tions. While,  under  the  limits  noted,  it  is  possible  to  construct 
500-volt  rotary  converters  for  60  cycles,  good  design  has  deter- 
mined that  25  cycles  is  the  proper  frequency  for  converter 
work  at  such  pressure. 

Since  the  direct  and  alternating  currents  flow  in  the  same 
windings,  revolving  in  the  same  field,  it  will  be  appreciated  that 
the  direct-current  voltage  bears  a  constant  ratio  to  the  alter- 
nating voltage;  the  maximum  value  of  the  internal  alternating 
e.m.f.  being  equal  to  the  direct  e.m.f.  The  effective  value 
of  the  alternating  e.m.f.  observed  will  depend  upon  the 
form  of  the  e.m.f.  wave  and  upon  the  points  on  the  windings 
between  which  the  voltage  is  taken.  Assuming  a  sine  wave 
and  denoting  the  direct  e.m.f.  by  1,  the  effective  alter- 
nating e.m.f.  is  about  0.71  for  two-phase  machines  and  star- 
connected  six  phasers,  and  about  0.61  for  three-phase  and  delta- 
connected  six-phase  machines.  The  wave  form,  and  hence  this 
ratio,  can  be  materially  changed  by  altering  the  slope  of  the 
pole-faces. 

EXCITATION  OF  SYNCHRONOUS  MACHINES. 

The  counter  e.m.f.  of  the  converter,  both  as  to  wave  form  and 
magnitude,  must  be  equal  to  that  of  the  supply  system.  If  the 
converter  does  not  tend  of  itself  to  produce  a  wave  similar 
and  equal  to  that  of  the  system,  corrective  currents  will  flow 
in  the  armature  windings,  which  currents  so  react  upon  the 
field  that  the  generated  e.m.f.  will  have  a  wave  form  exactly 
the  same  as  that  of  the  supply.  As  far  as  converter  output  is 
concerned,  these  corrective  currents  are  wattless.  They,  how- 
ever, affect  the  regulation  of  the  system  and  waste  energy  in 
the  resistance  of  the  connecting  circuits  and  should  therefore 
be  eliminated  when  possible. 

If  the  converter  is  excited  to  give  an  e.m.f.  less  than  that  of 
the  system  when  it  is  running  at  the  speed  at  which  the  alter- 
nations of  the  supply  designate  that  it  must  run,  a  lagging  cur- 
rent will  be  drawn  from  the  system,  which  current  tends  to 
strengthen  the  motor  field  so  that  the  generated  e.m.f.  is  made 
equal  to  that  of  the  system.  Similarly,  if  the  converter  field 
is  overexcited,  the  current  drawn  from  the  supply  mains  will 
be  leading  and  will  thus  demagnetize  the  field  sufficiently  to 


MOTORS  AND  CONVERTERS.  167 

make  the  e.m.f.  equal  to  that  of  the  system.  It  is  thus  plain 
that  the  current  demanded  depends  upon  the  field  excitation 
and  will  be  least  for  that  excitation  which  would  cause  the 
machine  to  generate  an  e.m.f.,  when  running  at  normal  speed, 
equal  to  that  of  the  system. 

HUNTING  OF  SYNCHRONOUS  MACHINES. 

The  mean  speed  of  the  converter  must  equal  the  mean  speed 
of  the  generator,  but  the  instantaneous  speeds  of  the  two  may 
be  quite  different,  as  will  be  seen  later.  The  synchronizing 
current,  which  holds  the  converter  in  step  with  the  system, 
tends  to  cause  the  converter  to  follow  any  irregularity  in  the 
frequency  of  the  supply-current. 

The  tendency  to  irregular  angular  velocity  in  each  revolution 
is  inherent  in  the  construction  of  reciprocating  engines  and  is 
augmented  by  the  periodic  hunting  of  the  governors  of  engines 
operating  alternators  in  parallel.  The  inertia  of  the  converter 
armature  causes  it  to  tend  to  run  at  a  constant  speed,  and  if 
the  alternations  of  the  supply  are  irregular,  during  a  portion 
of  the  time  the  converter  will  be  ahead  of  the  system,  and  at 
other  times  it  will  be  lagging  behind.  During  the  time  of  rela- 
tive phase  displacement  between  the  converter  armature  and 
the  system,  the  synchronizing  current  acts  to  draw  the  arma- 
ture into  perfect  step.  If  additional  forces  are  brought  to 
bear  upon  the  converter  during  the  period  of  phase  shifting, 
the  relative  oscillations  may  be  either  increased  or  decreased 
according  to  the  time-direction  of  such  forces. 

This  action  is  very  similar  to  the  swinging  of  a  pendulum. 
If,  when  the  swing  is  in  one  direction  there  is  given  it  an  im- 
pulse in  the  same  direction,  the  amplitude  of  the  swing  is  in- 
creased, and  if  the  impulse  is  given  in  the  opposite  direction, 
the  amplitude  is  diminished.  The  periodic  hunting  of  the 
engine  governor,  the  steam  admissions,  the  momentum  of  the 
reciprocating  parts,  the  inertia  of  the  generator  armature  and 
that  of  the  converter  armature,  are  elements  which  tend  to 
increase  or  diminish  this  oscillation. 

Much  theoretical  and  experimental  work  was  undergone  be- 
fore a  complete  cure  for  the  tendency  to  this  periodic  phase 
shifting  was  found.  Among  the  methods  at  present  used  may 
be  mentioned  the  heavy  flywheel  effect  for  the  converter,  and 


168  ALTERNATING  CURRENT  MOTORS. 

a  magnetically  weak  armature  compared  with  the  field.  In 
either  of  these  cases,  the  converter  armature  tends  to  revolve 
at  a  mean  speed  independent  of  the  relative  irregularity  of 
the  frequency  of  the  supply. 

The  method  which  has  been  found  most  satisfactory  for  the 
prevention  of  hunting  of  rotary  converters  is  the  use  of  damping 
devices.  These  are  usually  of  the  form  of  copper  shields  be- 
tween or  surrounding  the  poles,  often  covering  a  portion  of  the 
pole-tip  or  even  imbedded  in  the  pole  proper.  As  the  armature 
oscillates  back  and  forth  across  its  normal  position,  the  shifting 
armature  magnetism,  produced  by  the  unconverted  portion  of 
the  motor  current,  induces  current  in  the  low-resistance  copper 
shields,  which  current  always  opposes  the  shifting  magnetism 
producing  it.  The  damping  action  thus  brought  into  play 
when  the  field  is  suddenly  distorted  has  the  effect  of  suppressing 
the  oscillations.  When  the  alternations  of  the  supply  are  ir- 
regular, the  damping  devices  act  to  cause  the  converter  to 
tend  to  follow  the  irregularities,  but  prevent  an  exaggeration 
of  the  momentary  phase  displacement  of  the  armature  and  thus 
have  a  steadying  effect  upon  the  whole  system. 

STARTING  OF  SYNCHRONOUS  CONVERTERS. 

Before  a  rotary  can  be  placed  into  active  service,  it  must  be 
brought  up  to  synchronous  speed  and  into  step  with  the  supply 
system.  Methods  in  use  for  accomplishing  this  result  are  as 
follows: 

(1)  Since  polyphase  currents  are  universally  used  as  supply, 
the  application  of  the  alternating  currents  directly  to  the  sta- 
tionary armature  without  field  excitation  will  result  in  a  rotating 
magnetic  field  about  the  armature  core.  The  eddy  currents 
thereby  induced  in  the  pole-faces  will  exert  a  torque  on  the 
armature  and  cause  it  to  tend  to  speed  up  to  synchronism. 
Under  the  condition  of  starting,  the  step-up  transformer  rela- 
tion between  the  field  and  armature  windings  causes  a  rela- 
tively large  e.m.f.  to  be  generated  in  each  field  coil.  To  lessen 
danger  from  this  source,  the  windings  on  the  separate  poles 
may  be  isolated  from  each  other  so  that  the  e.m.f.  generated 
in  the  coils  will  not  be  in  the  normal  series  relations,  and  thus 
the  total  e.m.f.  across  any  two  points  may  be  limited  to  that 
generated  in  one  pole  winding  alone.  When  a  shunt  to  the 


MOTORS  AND  CONVERTERS.  169 

series  coils  is  used,  it  must  be  opened  at  starting,  otherwise  the 
heavy  alternating  current  sent  through  it  and  the  series  coils 
may  cause  excessive  heating. 

(2)  Where  the  station  equipment  will  permit,  the  converter 
may  be  started  up  as  a  direct-current  motor.     The  direct  cur- 
rent may  be  obtained  from  a  storage  battery,   from   another 
converter  or  a  motor-generator  set  may  be  installed  for  this 
purpose.     A  device  for  automatically  tripping  the  direct-current 
circuit-breaker  upon  closing  the  alternating-current  switch  has 
proved  a  valuable   addition   to  the   equipment  for  converters 
started  by  this  method 

(3)  A  method  extensively  employed  by  one  of  the  leading 
manufacturing  companies  is  the  use  of  separate  motors  for  starting 
one  or  more  of  the  converters  of  the  sub- station  equipment. 
The  motors  may   conveniently  be  of  the  induction  type  and 
therefore   started   by   standard   methods   for  this   purpose.     A 
common  location  for  the  induction  motor  secondary  is  upon 
an  extension  of  the  converter  shaft.     Since  the  induction  motor 
must  experience  a  slip  of  some  value,  it  is  necessary,  in  order 
to  bring  the  converter  to  full  synchronism,  for  the  motor  to 
have  a  less  number  of  magnet  poles  than  the  converter. 

P  COMPOUNDING  OF  SYNCHRONOUS  CONVERTERS. 

In  street  railway  and  similar  work  it  is  always  desirable  to 
increase  the  station  pressure  as  the  load  comes  on,  in  order 
that  the  line  voltage  shall  remain  more  nearly  constant.  The 
dependence  of  the  direct-current  voltage  of  the  converter  upon 
that  of  the  alternating  supply  has  been  commented  upon.  In 
order,  therefore,  to  increase  the  pressure  of  the  output  it  is 
necessary  to  increase  the  pressure  of  the  supply  also.  This 
increase  may  be  obtained  by  the  use  of  variable-ratio  step- 
down  transformers,  or  by  the  insertion  of  reactance  in  the 
supply  circuit  and  running  the  load  current  through  a  few 
turns  around  the  field  poles. 

We  have  found  previously  that  if  the  converter  is  over- 
excited, leading  currents  will  be  drawn  from  the  supply,  while 
if  the  excitation  is  below  normal,  lagging  currents  will  be  drawn. 
If  a  lagging  current  be  drawn  through  a  reactance,  the  collector 
ring  voltage  will  be  lowered.  If,  however,  leading  current  be 
drawn,  through  the  reactance  the  voltage  will  be  raised.  The 


170  ALTERNATING  CURRENT  MOTORS. 

change  in  the  phase  of  the  current  to  the  converter  is  governed, 
by  the  excitation,  which  is  in  turn  regulated  by  the  load  current, 
so  that,  with  series  reactance,  the  effect  of  the  series  coils  on 
the  field  of  the  converter  is  quite  similar  to  that  of  the  com- 
pounding on  the  ordinary  direct-current  railway  generator.  In 
operation  it  is  sometimes  found  that  the  transmission  line  and 
converter  circuits  possess  sufficient  self  induction  so  that  addi- 
tional reactance  is  unnecessary. 

INVERTED  CONVERTERS. 

The  rotary  converter  is  an  entirely  reversible  piece  of  appar- 
atus. If  fed  with  alternating  current  of  a  certain  voltage,  it 
will  supply  direct  current  of  a  corresponding  (not  equal)  volt- 
age, and  similarly,  if  fed  with  direct  current  it  will  deliver 
alternating  current  of  corresponding  voltage.  When  operated 
to  convert  from  direct  to  alternating  current,  the  rotary  is 
called  by  the  somewhat  ill-chosen  term  "  inverted  converter." 

When  driven  by  alternating  currents  its  speed  is  governed  by 
the  alternations  of  the  supply  quite  independent  of  all  other 
conditions.  When  run  from  the  direct-current  side,  however, 
its  speed  is  determined  by  the  relation  of  its  field  strength  and 
impressed  e.m.f.  at  the  brushes.  It  operates  in  this  respect 
exactly  like  a  direct-current  motor.  If  the  field  from  any 
cause  becomes  weakened,  the  converter  will  speed  up  until  its 
armature  conductors  cut  the  field  magnetism  at  a  rate  to  gen- 
erate an  e.m.f.  equal  to  the  internal  impressed  e.m.f.  If  the 
field  be  strengthened  the  speed  will  be  correspondingly  decreased. 
Obviously,  therefore,  the  frequency  of  the  alternating-current 
output  may  be  quite  irregular,  though  the  e.m.f.  be  constant, 
if  the  field  fluctuates  in  strength. 

As  far  as  the  alternating-current  output  is  concerned,  the  in- 
verted converter  operates  as  a  generator.  In  an  alternating- 
current  generator,  lagging  currents  weaken  the  field,  while 
leading  currents  have  the  opposite  effect.  With  constant  ex- 
ternal field  excitation,  the  running  strength  of  the  field  will, 
therefore,  depend  upon  the  character  of  the  alternating-current 
load.  When  used  to  supply  power  for  induction  motors  and 
similar  apparatus,  the  current  drawn  will  have  a  lagging  com- 
ponent which  weakens  the  field  and  tends  to  increased  speed. 
Safety  to  the  converter  and  motors  necessitates  that  the  increase 


MOTORS  AND  CONVERTERS.  171 

in  speed  be  limited,  while  satisfactory  service  requires  that  this 
tendency  be  counter-balanced. 

The  following  methods  are  in  use  for  overcoming  the  tendency 
to  irregular  speed: 

(1)  Magnetically  weak  armature  compared  with  the  field.     It 
is  possible  by  this  method  to  operate  a  converter  on  full  zero 
power  factor  current  without  very  materially  weakening  the  field. 

(2)  Separate  field  excitation  supplied  from  a  direct-current 
generator  driven  synchronously  with  the   converter.     Any  in- 
crease in  converter  speed  causes  the  exciter  generator  to  supply 
more   field  current   and  thus  counteracts  the  influence  of  the 
lagging    armature    current.     This    latter    arrangement    can    be 
made  to  regulate  for  very  small  variations  in  speed  by  operating 
the  exciter  field  below  saturation  and  the  converter  field  at  a 
high    magnetic    density    and    having    the    converter    armature 
relatively  magnetically  weak.     A  very  slight  increase  in  speed 
causes  a  large  increase  in  field  current,  while  at  the  same  time  the 
armature  current  has  a  small  demagnetizing  effect  upon  the  field. 

The  operation  of  a  rotary  converter  is  in  general  much  more 
satisfactory  than  that  of  a  corresponding  direct-current  gen- 
erator. This  is  due  to  several  causes:  (1)  Absence  of  field  dis- 
tortion; the  rotary  is  both  a  generator  and  a  motor.  As  a 
generator  the  armature  current  tends  to  distort  the  field  in  a 
direction  opposite  to  the  distortion  as  a  motor.  The  effects  of 
the  armature  reactions,  therefore,  neutralize  each  other,  and  since 
there  is  no  shifting  of  the  field,  the  point  of  commutation  does 
not  vary  with  the  load,  and  sparkless  commutation  results. 
(2)  Lessened  friction  loss.  (3)  Greater  output  from  same  arma- 
ture. Since  the  load  current  at  portions  of  each  revolution 
feeds  directly  from  the  alternating-current  side  without  travers- 
ing the  whole  winding,  as  must  be  the  case  with  a  generator,  the 
effective  armature  resistance  is  less  than  it  is  for  the  same 
armature  used  in  a  generator. 

PREDETERMINATION  OF  PERFORMANCE  OF  SYNCHRONOUS 
CONVERTERS. 

Due  to  the  simultaneous  operation  of  a  rotary  converter, 
both  as  a  motor  and  as  a  generator,  the  field  distortion  from  the 
motor  action  is  to  some  extent  counteracted  by  that  from  the 
generator  action,  as  has  just  been  stated,  so  that  under  proper 


172 


ALTERNATING  CURRENT  MOTORS 


field  excitation,  the  field  strength  remains  quite  approximately 
constant  throughout  a  great  range  of  load.  Hence,  the  armature 
iron  loss  varies  but  slightly  with  the  load  and,  with  a  degree 
of  accuracy  fairly  equivalent  to  that  obtaining  with  constant 
potential-transformers,  the  iron  loss  may  be  considered  to  be 
independent  of  the  load  current.  The  variable  loss  is  due  almost 
exclusively  to  the  copper  loss  in  the  armature  winding. 

The  rotary  converter  with  constant  impressed  alternating 
e.m.f.  considered  as  a  direct-current  generator,  tends  always  to 
produce  the  same  direct  external  e.m.f.  The  apparent  measur- 
able pressure,  however,  drops  off  as  the  load  is  applied,  due  to 


70  SO  90        100 

FIG.  87. — Characteristics  of  Rotary  Converter. 

the  copper  loss  of  the  armature,  and  such  drop  is  a  direct  measure 
of  the  loss  within  the  armature. 

At  any  chosen  value  of  load  current  the  sum  of  this  loss  in 
watts  added  to  the  output  watts  of  the  converter  gives  a  value 
which  would  be  directly  determined  by  the  product  of  the  direct 
e.m.f.  at  its  no-load  value  and  the  load  current  at  its  chosen 
value.  It  thus  appears  that  with  load  amperes  plotted  as  ab- 
scissas and  watts  as  ordinates  the  curve  of  armature  output 
plus  copper  loss  due  to  load  current  is  a  right  line  and  may  be 
drawn  at  once  for  any  value  of  output  current  (Fig.  87).  The 
ratio  of  the  watts  loss  in  the  armature  copper,  due  to  any  load 


MOTORS  AND  CONVERTERS.  173 

current,  to  the  value  of  the  load  current  gives  the  effective 
value  of  the  armature  resistance. 

Knowing  the  no-load  losses  of  the  converter  and  the  effective 
armature  resistance*  the  complete  performance  may  be  calcu- 
lated as  follows: 

Let  W  =  no-load  watts  input, 

R  =  effective  armature  resistance, 
E  =  no-load  direct  e.m.f., 
I  =  any  chosen  value  of  load  current; 
then        PR  =  copper  loss  of  armature  due  to  load, 
E  —  I  R  =  apparent  external  direct  e.m.f., 
W  +  EI  =  input, 
El-PR  =  output, 

El  -PR 

"=  efticiency> 


which  becomes  a  maximum  when  PR  =  W,  as  a  close  approx- 
imation. 

It  should  be  noted  that  the  losses  are  W  +  PR,  and  that  while 
the  ratio  of  E  I  to  W  +  PR  is  a  maximum  when  PR  =  W,  at 
any  armature  load  current,  7,  the  input  is  I  E  +  W  and  not 
simply  /  E. 

The  above  equations  are  based  on  the  assumption  of  constant 
iron,  friction  and  windage  loss,  which  assumption  is  closely 
exact  as  stated  above.  In  addition  to  these  losses,  the  value,  W 
includes  the  armature  copper  loss  for  the  no-load  current,  and 
the  field  copper  loss  for  exciting  current.  Since  for  efficient 
service  the  exciting  current  should  have  a  constant  value,  it 
follows  that  the  loss  from  this  source  decreases  as  the  machine 
is  loaded  due  to  the  fact  that  the  direct  voltage  decreases, 
requiring  less  loss  in  the  regulating  rheostat.  The  no-load 
armature  current  is  of  totally  an  alternating  nature  and  traverses 
the  whole  armature  winding,  and  during  a  portion  of  its  route 
through  the  armature  is  superposed  upon  that  part  of  the 
alternating  supply  current  which  is  about  to  be  converted  to 
direct  current.  While  its  effect  alone  upon  the  armature  re- 
sistance would  give  a  constant  value  of  loss,  when  the  two 
currents  intermingle  their  combined  loss  is  greater  than  the 
sum  of  the  losses  of  the  two  considered  separately,  since  in  any 
case  (x  +  y)2  is  greater  than  tf  +  y*.  It  is  thus  seen  that,  among 


174  ALTERNATING  CURRENT  MOTORS. 

the  losses  which  have  a  practically  constant  value,  one  in- 
creases with  the  load  while  another  decreases,  tending  somewhat 
to  keep  the  total  at  a  constant  value. 

From  the  above  facts  and  equations  it  appears  that  the  curves 
of  constant  losses,  variable  losses,  output,  input  and  efficiency  may  be 
constructed  from  the  two  value,  no-load  input  and  effective  arma- 
ture resistance. 

Fig.  87  gives  graphically  the  results  of  calculations  of  the 
characteristics  of  a  certain  rotary  converter  of  which  the  no-load 
losses  are  1000  watts  and  effective  armature  resistance  .125  ohm. 

SIX-PHASE  CONVERTERS. 

Due  to  the  fact  that  the  alternating  current  of  the  motor 
portion  of  the  converter  flows  in  general  in  a  direction  opposed 
to  that  of  the  direct-current  generator  portion,  the  effective 
armature  resistance  for  polyphase  converters  is  less  than  that 
of  the  same  machine  used  as  a  direct-current  generator.  The 
ratio  of  effective  armature  resistance  to  its  true  generator  value 
is  as  follows: 

2  rings  converter,  1.39 

3  "  "  .56 

4  "  "  .37 
6  "  "  .26 
8  M  "  .21 

The  ratio  of  effective  to  true  armature  resistance  depends 
upon  the  number  of  phases.  If  R  a  represents  the  true  arma- 
ture resistance,  and  R  the  effective  armature  resistance,  and  we 
assume  the  full-load  rating  of  the  machine  to  be  governed 
wholly  by  the  heating  of  the  armature  conductors,  then  the 
output  current  will  be  greater  as  a  rotary  converter  than  as  a 
generator  by  the  ratio, 


2 

\   R 


The  values  of  ^  and  ?  are  as  follows: 

K 


Ra  iRa 

~R          \Tf 


Three-phase  rotaries  ..........  1  .  80         1  .  34 

Quarter-phase    rotaries  ........  2  .  66         1  .  63 

Six-phase   rotaries  ............  3.  76         1  .  94 


MOTORS  AND  CONVERTERS.  175 

The  above  facts  bring  forward  another  of  equal  importance. 
It  is  evident  from  the  figures  just  given  that  an  armature  of  a 
converter  connected  up  six-phase  will  give  a  much  larger  output 
than  when  used  three-phase.  The  theoretical  ratio  is  about 
1  to  1.45.  In  practice  this  would  be  slightly  modified  by 
wattless  currents,  if  such  be  present. 

The  first  thought  of  the  use  of  six-phase  converters  suggests 
numerous  complications  of  connections,  which,  upon  further 
investigation,  are  found  not  to  exist.  With  reference  to  the 
converter  proper,  the  only  change  necessary  is  the  addition  of 
three  more  collector  rings  at  a  very  small  expense. 

An  examination  of  the  connections  of  a  six-phase  armature  will 
reveal  the  fact  that,  if  only  alternate  rings  be  considered,  ne- 
glecting for  the  moment  the  additional  three  rings,  we  have 
a  true  three-phase  armature.  Now  considering  only  the  other 
three  rings  alone,  we  have  again  a  true  three-phase  armature. 
Further  examination  will  show  that  at  any  given  instant  the 
e.m.f.  between  two  rings  of  one  set  chosen  as  above,  is  in  direct 
phase  opposition  to  the  e.m.f.  between  the  corresponding  two 
rings  of  the  other  set ;  this  is  true  of  each  pair  of  rings  of  each  set. 
We  therefore  find  that  the  three-phase  e.m.f.  in  one  set  is  dis- 
placed just  180  degrees  from  the  three-phase  e.m.f.  in  the  other 
set.  The  connections  to  obtain  six-phase  currents  from  the  two 
independent  three-phase  circuits  are  obvious  from  this  explana- 
tion. • 

SIX-PHASE  TRANSFORMATION. 

The  flexibility  of  polyphase  circuits  in  general  is  well  exem- 
plified by  the  numerous  interconnections  of  transformer  coils 
which  may  be  employed  to  produce  six  phases  from  two  or 
three  phases.  The  transformation  from  three  to  six  phases 
may  be  accomplished  by  the  use  of  three  transformers,  each 
having  one  secondary  connected  "  star  "  fashion,  or  by  three 
transformers,  each  having  two  secondaries,  connected  in  "  star," 
"  delta,"  or  "  ring  ";  or  by  the  use  of  two  transformers,  each 
with  two  secondaries,  connected  in  "  delta  "  or  "  tee  ";  or  two 
transformers  each  with  one  secondary  may  be  used  as  com- 
bined autotransformers  and  transformers  to  obtain  the  desired 
conversion.  A  few  of  the  methods  of  transformation  just  men- 
tioned are  of  interest  only  from  an  academic  point  of  view,  and 
such  will  not  be  further  discussed;  only  those  which  possess 


176 


ALTERNATING  CURRENT  MOTORS. 


points  of  special  interest  or  are  of  practical  value  will  be  con- 
sidered in  detail. 

In  many  respects  a  six-phase  system  may  be  represented  as 


Tvo-] 


df— 


VUiMMA^ 

f? 

L 

•mer  A 

Q&CQQ&n 

1 

1 

r^ 

a  ' 

"*Lase 
uit 

4 
1 

i 

i 

5T                                                   d 

X^V, 

^.--61.22-V_--^ 

i  '"v   ; 

1         V    § 

^          >ai 

Six-Phase 

Keceiver 


5  E.M.F.  Correspond-  to          I 
i  100  V.  D.C.  Syn.  Coarerttr  $ 


FIG.   88. — Two-phase   to   Six-phase  Transformation;   Two 
Transformers,  Tee  Secondary. 

two  superposed  three-phase  systems,  and  a  certain  degree  of 
simplicity  in  tracing  the  transformation  circuits  may  be  ob- 
tained by  keeping  this  fact  in  mind.  This  fact  is  somewhat 


.-61.22-V J 


FIG.  89. — Three-phase  to  Six-phase  Transformation;  Two 
Transformers,  Tee  Secondary. 

emphasized  by  the  method  of  transforming  from  two  to  six 
phases  that  is  shown  by  Fig.  88.  As  will  be  observed,  the  two 
transformers  are  wound  quite  similarly  to  those  used  in  the 


MOTORS  AND  CONVERTERS. 


177 


Scott  method  of  transforming  from  two  to  three  phases;  the 
difference  being  in  the  division  of  each  secondary  winding  into 
two  parts.  The  six  secondary  coils  are  connected  so  as  to  form 
two  three-phase  systems.  These  two  systems  are  twice  re- 
versed with  reference  to  each  other;  electrica  ly  at  the  trans- 
formers, and  mechanically  at  the  six-phase  receiver,  so  that  the 
separate  tendencies  to  motion  would  be  in  the  same  direction 
of  rotation.  Since  there  exists  no  interconnection  between  the 
two  three-phase  circuits  the  cross  e.m.fs.  shown  in  Fig.  88  can 


©^ — 61.22V — ^2) 

J8  3?^. 


FIGS.  90-93. — Symmetrical  Voltage  Diagrams. 

be  observed  only  when  the  six-phase  receiver  of  itself  tends  tc 
produce  such  e.m.fs. 

As  a  comparison  between  Fig.  88  and  Fig.  89  will  show,  a 
relatively  slight  change  in  the  primary  coil  of  one  transformer 
renders  the  two-phase  to  six-phase  connections  of  circuits  ap- 
plicable to  transformation  from  three  to  six  phases.  The  re- 
sistances shown  in  Fig.  89  are  not  essential  to  the  operation  of  a 
six-phase  rotary  converter  or  similar  apparatus,  though  when 
such  apparatus  is  absent  the  existence  of  e.m.fs.  in  six-phase 
relation  can  best  be  proved  by  the  use  of  a  voltmeter  when  re- 


178 


ALTERNATING  CURRENT  MOTORS. 


sistance  is  thus  used.  By  varying  the  points  on  the  separate 
resistances  which  are  joined  together  a  variety  of  superposed 
three-phase  e.m.fs.  may  be  obtained,  the  existence  of  which 
may  be  proved  by  a  voltmeter.  Figs.  90  to  93  indicate  a  few 


FIG.  94. — Three-phase  to  Six-phase  Transformation;  Delta 
Primary,  Star  Secondary. 

of  the  symmetrical  figures  thus  produced.  Under  operating 
conditions  the  generator  action  of  a  six-phase  rotary  converter 
causes  the  e.m.fs.  to  assume  the  relation  shown  by  Fig.  91, 
and  the  use  of  resistance  for  such  purpose  is  entirely  superfluous. 


FIG.  95. — Three-phase  to  Six-phase  Transformation;  Delta 
Primary,  Delta  Secondary. 

Perhaps  of  the  many  methods  for  transforming  from  three  to 
six  phases,  the  one  possessing  the  greatest  simplicity  in  trans- 
former circuits  is  that  in  which  the  secondaries  are  star  con- 
nected, and  the  primaries  either  star  or  delta  connected.  Fig. 


MOTORS  AND  CONVERTERS. 


179 


94  shows  such  interconnection  of  circuits  with  the  primaries 
connected  in  delta.  With  the  six-phase  receiver  absent,  a  volt- 
meter would  register  only  three  separate  single-phase  e.m.fs., 
and  would  indicate  no  cross  e.m.fs.  between  the  phases.  By 
tapping  each  secondary  coil  at  its  middle  point,  and  joining 
these  three  points  to  form  a  common  neutral,  all  the  e.m.fs.  of 
the  symmetrical  six-phase  receiver  will  be  properly  indicated 
on  a  voltmeter.  As  stated  previously,  however,  the  operation 
of  a  six-phase  receiver  does  not  depend  upon  the  production 
of  the  symmetrical  figure  external  to  the  receiver  and  the  per- 
formance will  be  quite  satisfactory  without  joining  the  three 
neutral  points  of  the  separate  single-phase  circuits. 

Fig.   95  indicates  connecting  circuits  for  three-phase  to  six- 


FIG.  96. — Three-phase  to  Six-phase  Transformation;  Delta 
Primary,  Ring  Secondary. 

phase  transtormation ,  both  primary  and  secondary  coils  being 
connected  in  delta.  No  change  whatever  need  be  made  in  the 
connections  of  the  secondary  circuits  in  order  to  operate  the 
primary  coils  in  star,  though  the  e.m.f.  per  primary  coil  would 
thereby  need  to  be  decreased  in  the  ratio  of  \/3  to  1,  of  course. 
A  comparison  of  Fig.  95  and  Fig.  96  will  reveal  the  fact  that 
the  same  transformers  may  be  used  for  either  delta  or  ring 
connected  secondaries,  though  a  change  in  the  ratio  of  primary 
to  secondary  turns  per  coil  would  be  necessary  in  order  to 
operate  the  receiver  at  the  same  e.m.f.  for  the  two  methods  of 
transformation;  the  change  being  as  the  ratio  of  the  side  of  an 
equilateral  triangle  to  that  of  a  regular  hexagon  inscribed 
within  the  same  circle. 


180  ALTERNATING  CURRENT  MOTORS. 

RELATIVE  ADVANTAGES  OF  DELTA  AND  STAR-CONNECTED  PRI- 
MARIES. 

Since,  when  three  transformers  are  connected  in  delta,  one 
may  be  removed  without  interrupting  the  performance  of  the 
circuit — the  other  two  transformers  in  a  manner  acting  in  series 
to  carry  the  load  of  the  missing  transformers — the  desire  to 
obtain  immunity  from  a  shut-down  due  to  the  disabling  of  one 
transformer  has  led  to  the  extensive  use  of  the  delta  connection 
of  transformers  especially  on  the  low  potential  six-phase  side. 
It  is  to  be  noted  in  this  connection  that  in  case  one  transformer 
is  crippled  the  other  two  will  be  subjected  to  greatly  increased 
losses.  If  three  delta-connected  transformers  be  equally  loaded 
until  each  carries  100  amperes,  there  will  be  173  amperes  in 
each  external  circuit  wire.  If  one  transformer  be  now  removed 
and  173  amperes  continues  to  be  supplied  to  each  external  circuit 
wire,  each  of  the  remaining  transformers  must  carry  173  amperes, 
since  it  is  now  in  series  with  an  external  circuit.  Therefore,  each 
transformer  must  now  show  three  times  as  much  copper  loss  as 
when  all  three  transformers  were  active,  or  the  total  copper  loss  is 
now  increased  to  a  value  of  six  relative  to  its  former  value  of  three. 

A  change  from  delta  to  star  in  the  primary  circuit  alters  the 
ratio  of  the  transmission  e.m.f.  to  the  receiver  e.m.f.  from  1  to 
X/jf  On  account  of  this  fact,  when  the  e.m.f.  of  the  transmission 
circuit  is  so  high  that  the  successful  insulation  of  transformer  coils 
becomes  of  constructive  and  pecuniary  importance,  the  three- 
phase  line  side  of  the  transformers  is  frequently  connected  in  star. 

When  rotary  converters  are  employed  for  supplying  power 
to  lighting  circuits,  it  is  frequently  desirable  that  a  lead  at 
neutral  potential  be  run  on  the  direct-current  side  of  the  sys- 
tem. Since  the  input  side  of  the  converter  is  electrically  con- 
nected to  the  output  side,  it  follows  that  the  neutral  point  on 
the  alternating-current  end  of  the  converter  is  simultaneously 
the  neutral  point  on  the  direct-current  end.  For  this  reason, 
it  is  sometimes  advantageous  to  join  the  low-potential  windings 
of  the  transformers  in  such  a  manner  as  to  allow  an  electrical 
connection  to  be  made  to  the  neutral  point  from  the  neutral 
conductor  of  the  three-wire  direct-current  system.  Thus  for  a 
three-ring  converter  the  coils  could  be  joined  in  "  tee  "  or  in 
"  star,"  for  a  four-ring  converter  the  coils  could  be  intercon- 
nected at  the  central  points,  while  for  a  six-ring  converter  the 
coils  could  be  arranged  in  double  interconnected  "  tee  "  or  "  star." 


CHAPTER  XII. 
ELECTROMAGNETIC  TORQUE. 

COMMUTATOR  MOTORS. 

Before  investigating  the  characteristics  of  single-phase  com- 
mutator motors,  it  is  well  to  review  a  few  facts  relating  to  the 
production  of  torque  by  electromagnetic  action,  and  to  ascer- 
tain some  method  by  which  rotative  torque  can  be  measured 
most  conveniently. 

If  a  bipolar,  direct-current  armature  be  placed  within  core 
material  having  uniform  magnetic  reluctance  around  the  air- 
gap,  as  for  example  within  an  induction  motor  stator,  and 
brushes  placed  upon  the  commutator  in  mechanical  quad- 
rature, as  shown  in  Fig.  97,  be  caused  to  carry  direct  current 
from  two  isolated  sources  of  supply,  it  will  be  found  that  the 
armature  has  no  tendency  to  motion  in  either  direction,  what- 
soever may  be  the  values  of  the  two  currents  Upon  super- 
ficial examination  one  is  inclined  to  attribute  the  lack  of  torque 
to  the  fact  that  such  flux  as  may  be  in  mechanical  position  to 
give  force  by  its  product  with  any  current  existing  in  the  arma- 
ture is  caused  by  currents  in  the  same  armature,  and  the  two 
currents,  being  in  the  same  mechanical  structure,  could  not 
cause  motion  with  reference  to  any  external  body.  It  will  be 
evident  that  each  current  is  in  position  to  produce  force  in  a 
certain  direction  due  to  the  presence  of  the  flux  caused  by  the 
other  current.  It  is  not  immediately  apparent,  however,  that 
each  component  force  thus  produced  tends  to  give  motion  to  the 
armature  with  reference  to  the  stator,  and  that  the  cause  for 
the  lack  of  resultant  torque  is  the  opposition  in  direction,  with 
equality  in  value,  of  the  component  forces. 

EQUALITY  OF  TORQUES  FOR  UNIFORM  RELUCTANCE. 

From  the  fundamental  law  of  physics  that  a  force  of  one 
dyne  is  exerted  upon  each  centimeter  of  length  of  a  conductor 
per  unit  current  per  line  cf  force  flux  density  in  the  area  through 

181 


182 


ALTERNATING  CURRENT  MOTORS. 


which  the  conductor  passes,  is  obtained  the  torque  equation 
for  the  current  through  brushes  A  A  (Fig.  97),  due  to  the  pres- 
ence of  the  flux  caused  by  the  current  through  the  brushes  B  B, 

Ta  =  K  Ia  4>b 

where  K  is  a  proportionality  constant  depending  for  its  value 
upon  the  number  and  arrangement  of  the  armature  conductors. 
Similarly,  the  torque  for  the  current  through  the  brushes  B  B 
will  be 


the  constant,  K,  having  the  same  value  as  above. 


'FiG.   97. — Superposed   Direct-currents   in   Armature;    Two 
Fields  in  Mechanical  Quadrature;  No  Resultant  Torque. 

A  study  of  the  circuits  and  magnets  of  Fig.  97  will  show  that 
these  two  torques  are  opposite  in  direction,  so  that  the  resultant 
torque  is 

T-    Ta-Tb   =   K(Ia^b-Ib^a). 

With  uniform  reluctance  in  all  directions  across  the  air-gap 
and  through  the  core  material,  the  flux  per  unit  current  will 
be  the  same  in  both  axial  brush  lines,  so  that 


from  which  is  obtained   T  =  0. 


ELECTROMAGNETIC  TORQUE. 


183 


Hence,  under  the  conditions  assumed,  the  resultant  torque 
has  zero  value,  though  each  component  torque  may  have  a 
certain  definite  value  tending  to  give  motion  to  the  armature. 

INEQUALITY  OF  TORQUES  FOR  NON-UNIFORM  RELUCTANCE. 

If  the  reluctance  be  greater  in  the  axial  line  of  one  set  of 
brushes  than  in  that  of  the  other,  then  the  proportionality  be- 
tween the  current  and  the  flux  produced  thereby  becomes 
altered,  so  that  Ia^>b  n°  longer  equals  Ib<!>a,  and  the  resultant 
torque  assumes  a  value  proportional  to  their  difference.  A 


FIG.   98. — Superposed   Direct-currents   in   Armature;    One 
Quadrature  Field  Neutralized;  Good  Operating  Torque. 

change  in  the  relative  reluctance  in  the  two  directions  may  be 
obtained  by  removing  a  portion  of  the  core  material  in  one 
axial  brush  line,  thereby  retaining  the  projecting  poles  common 
in  direct-current  practice. 

Fig.  98  shows  a  method  by  which  the  flux,  which  current 
through  the  brushes  A  A  would  tend  to  produce,  may  be  ren- 
dered of  zero  value  for  any  amount  of  current  in  the  circuit, 
thus  giving  the  effect  of  infinite  reluctance  in  this  axial  brush  line. 
When  the  effective  turns  on  the  stator  core  are  equal  in  number 
to  those  on  the  armature,  with  circuits  connected  as  here  indi- 


184  ALTERNATING  CURRENT  MOTORS 

cated,  the  machine  will  operate  as  a  separately  excited  direct- 
current  motor.  The  torque  will  be  of  a  value  determined 
wholly  by  the  product  of  the  current  through  A  A  and  the 
flux  along  B  B,  and  will  be  in  no  wise  influenced  by  the  fact 
that  the  flux  is  produced  by  current  in  the  armature.  The 
statement  here  made  easily  admits  of  experimental  verification. 
While  the  facts  presented  above  are  more  of  theoretical  in- 
terest than  of  practical  importance  with  reference  to  direct- 
current  machinery,  they  form  the  essential  groundwork  upon 
which  are  based  the  fundamental  equations  for  determining 
the  characteristics  of  numerous  types  of  alternating-current 
commutator  motors,  now  being  developed. 

DETERMINATION  OF  TORQUE  BY  CALCULATION  OF  THE  OUTPUT. 

A  little  experience  with  the  well-known  mechanical  and  elec- 
trical methods  for  determining  torque  convinces  one  that  the 
latter  method  is  far  preferable  to  the  former  with  reference  to 
ease  of  adjustment,  flexibility  of  operation  and  reliability  of 
results.  For  ascertaining  the  output  from  either  mechanical  or 
electrical  motors,  perhaps,  the  most  fa"miliar  method  is  one  which 
involves  the  use  of  a  direct-current  generator,  of  which  the  sum 
of  the  input  and  transmission  losses  is  taken  as  the  value  of 
the  output  desired.  The  input  to  the  direct-current  generator 
is  found  as  the  sum  of  its  output  and  its  internal  losses.  In 
order  to  determine  the  internal  losses  of  the  generator,  it  is 
necessary  to  find  the  value  of  the  individual  iron,  friction  and 
copper  losses.  When  the  resistances  of  the  separate  circuits  of 
the  generator  and  the  currents  flowing  there  through  are  known, 
the  copper  losses  may  readily  be  calculated.  The  armature  iron 
loss  varies  both  with  the  speed  and  the  density  of  magnetism. 
That  the  effect  of  any  change  in  the  latter  may  be  eliminated, 
it  is  usual  to  operate  the  generator  as  a  shunt-wound  machine 
with  constant  field  excitation  and  with  the  armature  brushes 
at  the  mechanical  neutral  point,  under  which  conditions  the 
iron  loss  will  vary  at  a  rate  but  sligthly  greater  than  the  first 
power  of  the  speed,  and,  where  the  nature  of  the  test  so  dic- 
tates, the  value  may  accurately  be  determined  throughout  any 
desired  range  of  speed. 

In  cases  where  the  load  generator  and  the  driving  motor  are 
constructed  for  the  same  e.m.f.  and  capacity,  the  output  from 


ELECTROMAGNETIC  TORQUE. 


185 


the  generator  may  be  fed  back  into  the  supply  line,  the  test 
thereby  using  only  that  amount  of  power  necessary  to  overcome 
the  losses  of  the  two  machines.  In  these  latter  cases  the  in- 
dividual losses  of  each  machine  are  calculated  as  formerly  and 
each  is  subject  to  the  same  errors  as  before,  but  the  sum  of  the 
losses,  being  directly  measured,  is  accurately  determined  and 
may  be  used  as  a  check  on  the  separate  losses. 

The  method  given  below  combines  the  convenience  and  econ- 
omy of  the  "  loading  back  "  method,  is  subject  to  a  less  number 
of  sources  of  errors  and  is  applicable  to  all  types  of  motors, 
either  direct  or  alternating  current,  which  may  possess  either 
the  series  or  shunt  motor  characteristics,  and  in  many  cases 


Rheostat 


FIG.  99. — Determination  of  Torque. 
•^ 

it  may  with  equally  desirable  results  be  applied  to  the  testing 
of  either  mechanical  or  electrical  motors. 

MEASUREMENT  OF  TORQUE  BY  THE  LOADING  BACK  METHOD. 

The  circuit  diagram  of  Fig.  99  will  serve  to  make  clear  the 
method  of  connecting  the  apparatus  for  the  test  and  may  be 
used  to  explain  the  theory  upon  which  the  test  depends.  In 
Fig.  99  the  load  generator  is  shown  as  a  constant-potential, 
shunt-wound,  direct-current  machine,  while  the  driving  motor, 
as  shown,  is  a  series-wound  machine,  and  may  be  of  either  the 
direct  or  alternating-current  type.  It  is  desired  to  find  the 
torque  of  the  series  machine  at  various  speeds.  If  the  shunt 
machine  be  operated  as  a  motor — being  belted  to  the  series 


186  ALTERNATING  CURRENT  MOTORS. 

machine  which  is  run,  with  circuit  switch  C  open,  at  a  speed 
somewhat  below  that  at  which  the  value  of  the  torque  is  desired 
to  be  obtained — it  will  require  a  certain  armature  current,  70, 
at  a  certain  impressed  e.m.f.,  E.  If  now  the  switch,  C,  in  the 
circuit  to  the  series  machine,  be  closed,  the  shunt  machine 
will  be  driven  at  an  increased  speed  and  will  require  an  armature 
current  smaller  than  before — perhaps  of  negative  value — due 
to  the  accelerating  torque  transmitted  to  the  belt,  and,  if  the 
e.m.f.,  E,  and  the  field  current  of  the  shunt  machine,  remain 
constant,  the  value  of  the  torque,  exerted  by  the  series  machine, 
expressed  in  equivalent  watts  per  revolution  per  minute,  will  be: 

(70-7L)£ 
5 

where  70  is  amp.   taken  by  armature  of  shunt  machine  with 

switch,  C,  open; 

7L  is  amp.  taken  by  armature  of  shunt  machine  with  switch,  C, 

closed ; 

and  S  is  the  "  synchronous"  speed  of  the  set,  as  determined  by 

the  relation  of  the  e.m.f.  and  field  strength  of  the  shunt  machine. 

The  equation  above  expresses  the  value  of  the  torque  by  which 
the  series  machine  assists  the  shunt-wound  machine,  and  gives 
the  true  value  of  the  torque  which  the  series  machine  delivers 
to  its  own  shaft. 

The  convenience  of  this  method  in  comparison  with  one  which 
uses  the  shunt  machine  as  a  generator  will  be  appreciated  when 
it  is  considered  that  no  account  need  be  taken  of  the  internal 
losses  or  of  the  output  of  the  machine  and  that  the  field  current 
of  the  shunt  machine  and  therewith  the  speed  of  the  set  may 
be  adjusted  to  any  desired  value  for  each  determination  of 
torque  without  affecting  the  results.  The  economy  of  the  method 
is  due  to  the  fact  that  the  set  dissipates  only  that  amount  of 
power  represented  by  the  losses  of  the  two  machines,  all  excess 
of  power  being  returned  through  the  constant  potential  supply 
circuit  by  means  of  the  current  produced  by  the  generator 
action  of  the  shunt  machine. 

The  accuracy  of  the  method  depends  upon  the  following  facts: 
A  direct-current  motor  runs  at  a  speed  such  as  to  generate  an 
e.m.f.  less  than  the  impressed  by  an  amount  sufficient  to  force 
through  the  resistance  of  its  armature  a  current  of  a  value 


ELECTROMAGNETIC  TORQUE.  187 

such  that  its  product  with  the  field  magnetism  gives  the  torque 
demanded  at  its  shaft.  With  constant  field  magnetism  the 
electrical  torque  of  a  direct-current  machine,  operated  as  either 
a  motor  or  a  generator,  is  given  by  the  expression: 


where  I  is  the  armature  current, 
E  is  the  impressed  e.m.f., 

and  S  is  that  speed  at  which  the  counter  e.m.f.  of  rotation 
of  the  armature  windings  in  the  field  magnetism  equals 
the    impressed    e.m.f.;    that    is,    the    "synchronous" 
speed  as  used  above. 
If  Wo  =  watts  output  (electrical), 

R  =  resistance  of  armature, 
r.p.m.  =  actual  speed  of  armature, 

Wo         IE  -PR         I(E-IR)       I  EC 

then  D  =  -       ~  =  -        -  =  —  ^—  --  '  =  -  -  -  ,  where 
r.p.m.  r.p.m.  r.p.m.  r.p.m. 

EC        E 

EC  is  the  counter  e.m.f.  of  rotation.     But  -  —  =  —  ,   for  con- 

r.p.m.      S 

I  E 
stant  field  strength;  hence,  D  =  —  -,  as  given  above. 

kJ 

Since  for  a  certain  impressed  e.m.f.  S  has  a  definite  fixed 
value  for  each  adjustment  of  field  strength,  with  constant  field 
magnetism,  the  internal  electrical  torque  of  the  shunt  machine 
varies  directly  with  the  armature  current,  and  any  change  in 
the  value  of  this  current  serves  at  once  as  a  measure  of  the 
change  in  torque  -exerted  by  the  shunt  machine,  quite  inde- 
pendent of  all  other  conditions. 

ELIMINATION  OF  ERRORS. 

It  remains  now  to  show  why  the  change  in  torque  of  the 
shunt  machine  may  be  used  to  determine  the  torque  exerted 
by  the  driving  motor.  The  torque  delivered  to  the  shaft  of 
the  series  motor  is  less  than  the  internal  electrical  torque  of  the 
shunt  machine  by  that  necessary  to  overcome  the  iron  and 
friction  losses  of  the  shunt  machine  and  the  transmission  losses 
in  the  belt.  Since  the  belt  and  friction  losses  vary  directly 


188  ALTERNATING  CURRENT  MOTORS. 

with  the  speed,  it  will  be  evident  that  the  counter  torque  due 
thereto  will  be  constant.  For  constant  field  magnetism,  the 
armature  hysteresis  loss  varies  as  the  first  power,  and  the  eddy 
current  loss  as  the  square,  of  the  speed.  Since  in  comparison 
with  the  other  loss,  that  due  to  the  eddy  currents  is  relatively 
small,  the  sum  of  the  iron,  friction  and  transmission  losses  varies 
at  a  rate  inappreciably  greater  than  the  first  power  of  the  speed 
and  the  torque  necessary  to  overcome  these  losses  may,  for  prac- 
tical purposes,  be  taken  as  being  independent  of  the  slight  change 
in  speed.  The  change  in  the  internal  electrical  torque  of  the 
shunt  machine,  when  switch  C,  of  Fig.  99,  is  closed,  gives  at 
once  the  value  of  the  torque  delivered  to  its  shaft  by  the  series 
motor. 

The  method  outlined  above  may  be  used  to  determine  the 
torque  exerted  by  a  machine  when  such  torque  is  much  less  than 
that  necessary  to  drive  a  generator  of  any  capacity  whatsoever 
and  is,  therefore,  especially  advantageous  for  tests  where  it  is 
desired  to  find  the  torque,  at  high  speeds  of  machines  possessing 
series  motor  characteristics. 

In  Fig.  99  is  shown  a  series-wound  driving  motor,  but  it  will 
be  evident  that  the  change  in  torque,  as  given  by  the  variation 
of  the  current  taken  by  the  shunt  machine,  may  be  produced 
by  any  type  of  motor.  A  little  consideration  will  show  that  since 
at  any  given  speed,  the  torque  exerted  by  the  shunt  machine 
of  Fig.  99  may  be  adjusted  throughout  any  desired  range  by 
use  of  the  field  rheostat,  the  method  may  conveniently  be  ap- 
plied to  motors  possessing  practically  constant  load  speed  char- 
acteristics, such  as  those  of  the  direct-current,  shunt-wound 
type  or  of  the  alternating-current  induction  type,  and  that 
alternating-current  synchronous  motors  may  be  similarly 
tested,  if  after  adjustment  of  the  load  on  the  synchronous  motor 
by  means  of  the  rheostat  in  the  field  circuit  of  the  shunt  machine 
the  supply  of  electric  power  be  cut  off  from  the  synchronous 
machine  in  order  to  obtain  the  change  in  torque  exerted  by  the 
shunt  machine. 


CHAPTER  XIII. 

SIMPLIFIED   TREATMENT  OF  SINGLE-PHASE  COMMUTATOR 

MOTORS. 

THE  REPULSION  MOTOR. 

Mention  has  already  been  made  of  the  use  of  a  commutator 
on  the  revolving  secondary  of  a  single-phase  motor  for  the 
purpose  of  giving  to  the  rotor  a  starting  torque.  It  seems 
desirable  to  treat  this  so-called  repulsion  motor  more  in  detail 
and  to  explain  its  operation  more  fully.  The  verbal  descriptive 
matter  presented  below  will  serve  to  give  to  the  reader  a  fair 
idea  of  the  operating  characteristics  of  the  machine,  after  which 
the  more  complete  analytical  study  of  the  motor  may  be  under- 
taken. Since  the  mathematical  treatment  of  the  other  types 
of  commutator  motors  is  quite  the  same  as  that  used  with  the 
repulsion  motor,  it  is  believed  that  a  little  familiarity  in  the 
performance  of  the  repulsion  motor  will  be  of  great  assistance 
in  becoming  acquainted  with  the  characteristics  of  the  other 
machines. 

The  repulsion  motor  is  a  transformer,  the  secondary  core  of 
which  is  movable  with  respect  to  the  primary,  and  the  secondary 
coil  of  which  remains  at  all  times  short-circuited  in  a  line  in- 
clined at  a  certain  angle  with  the  primary  coil.  Such  a  machine 
is  represented  diagrammatically  in  Fig.  100.  Superficially  con- 
sidered, current  which  flows  in  the  secondary  (the  armature) 
by  way  of  the  brushes  acts  upon  the  field  produced  by  the 
primary  current  to  give  the  armature  a  torque  which  retains 
its  direction  with  the  simultaneous  reversal  of  the  two  currents. 
If  the  brushes  were  placed  in  line  with  the  field  poles,  maximum 
current  would  be  produced  in  the  secondary,  but  it  would 
have  no  tendency  to  move  because  such  torques  as  are  produced 
on  one  side  of  the  armature  would  be  opposed  by  those  produced 
on  the  other.  Similarly,  if  the  brushes  were  placed  at  right 
angles  to  the  field  axis  no  torque  would  be  obtained,  for  no 
current  would  flow  in  the  secondary.  A  further  analysis  will 

189 


190 


ALTERNATING  CURRENT  MOTORS. 


show  that  the  torque  depends  directly  upon  the  product  of  the 
secondary  (armature)  current  and  that  part  of  the  field  magnet- 
ism which  is  in  mechanical  quadrature  with  the  radial  line 
joining  the  secondary  brushes  and  which  is  in  time-phase  with 
the  armature  current.  In  fact,  the  torquo  follows  a  law  similar 
in  all  respects  to  that  which  holds  for  direct-current  machines. 

ELECTRIC  AND  MAGNETIC  CIRCUITS  OF  IDEAL  MOTOR. 

For  the  purpose  of  analysis,  it  is  convenient  to  divide  the 
primary  magnetism  into  two  components,   one  in  mechanical 


FIG.  100. — Circuits  of  Repulsion  Motor. 

quadrature  with  the  line  of  the  brushes,  to  give  the  torque, 
and  one  directly  in  line  with  the  brushes,  which  produces  cur- 
rent in  the  secondary  by  transformer  action.  In  commercial 
repulsion  motors  the  primary  winding  is  distributed  over  an 
approximately  uniformly  slotted  core  without  the  projecting 
poles  shown  in  Fig.  100,  and  the  assumption  is  made  that  at  any 
given  angle,  a,  to  which  the  brushes  are  shifted  from  the  field 
line,  the  flux  component  in  line  with  the  brushes  is  (f>  cos  a,  and 
that  in  quadrature  is  </>  sin  a,  where  <f>  is  the  total  primary  flux. 


SINGLE-PHASE  COMMUTATOR  MOTORS. 


191 


This  assumption  is  more  or  less  justified  by  the  fact  that  when 
no  current  flows  in  the  secondary,  such  component  values  of 
fluxes  give  a  resultant  equal  to  the  primary  flux  and  having 
the  proper  mechanical  position  on  the  core.  As  will  appear 
later,  however,  this  assumption  leads  to  error  in  assigning 
values  to  the  two  flux  components.  In  order  to  eliminate  all 
trouble  from  this  source  and  to  allow  the  conditions  to  be  clearly 
presented,  it  is  well  to  divide  the  primary  winding  up  into  two 
parts  placed  upon  two  projecting  cores  in  mechanical  quadrature 
one  with  the  other,  and  to  locate  the  brushes  in  line  with  one 
core,  as  shown  in  Fig.  101.  As  the  simplest  possible  case,  it 


FIG.  101. — Circuits  of  Ideal  Repulsion  Motor. 

will  be  assumed  that  the  number  of  turns  on  one  core  is  the  same 
as  on  the  other  and  equal  to  the  effective  turns  on  the  armature. 
Under  the  conditions  assumed,  when  no  current  flows  in  the  sec- 
ondary, the  primary  field  would  have  a  resultant  located  45° 
from  the  line  joining  the  two  brushes. 

If  the  secondary  brushes  be  connected  together  while  the 
rotor  is  stationary,  the  transformer  action  of  the  flux  in  A  will 
cause  a  current  to  flow  in  the  secondary,  which  current  tends 
to  reduce  the  flux  in  A  and  allow  more  current  to  flow  in  the 
primary  coil  and  to  increase  the  flux  in  B.  If  the  transformer 


192  ALTERNATING  CURRENT  MOTORS. 

action  were  perfect,  no  flux  would  remain  in  A,  while  the  flux 
in  B  would  assume  double  value  and  the  current  in  the  secondary 
would  equal  that  in  the  primary  (the  primary  and  secondary 
turns  at  A  being  equal).  Such  a  condition  would  exist  if  the 
primary  and  secondary  coils  were  devoid  of  resistance  and  local 
reactance  (magnetic  leakage). 

If  the  resistance  and  local  reactance  at  A  be  considered  negligi- 
ble, when  the  rotor  is  stationary  the  e.m.f.  across  the  coil  on  the 
core  A  will  be  zero,  while  that  on  B  will  be  equal  to  the  total 
impressed  e.m.f.  Assume  that  the  core  material  at  B  and  through 
the  armature  is  such  that  the  flux  produced  is  in  time  phase 
with  the  current  in  the  coil  and  proportional  at  all  times  to 
such  current;  that  is  to  say,  that  the  reluctance  of  the  magnetic 
path  of  B  is  constant.  Let  it  be  further  assumed  that  the 
reluctance  of  the  magnetic  circuit  of  A  is  equal  to  that  of  B. 
It  should  be  noted  that  these  assumptions  are  equivalent  in 
all  respects  to  those  which  are  invariably  implied  in  a  math- 
ematical or  graphical  treatment  where  the  coefficient  of  self- 
induction,  L,  is  taken  as  constant. 

PRODUCTION  OF  ROTOR  TORQUE. 

The  production  of  torque  at  the  rotor  can  now  be  investigated. 
The  flux  in  B  is  in  time  phase  with  the  primary  current,  while 
the  current  in  the  armature  is  in  phase  opposition  to  that  in 
the  coil  on  A.  Therefore,  the  armature  current  is  in  time 
phase  with  the  field  magnetism,  and  the  torque  (the  product 
of  the  two)  retains  its  sign  as  the  two  reverse  together.  If  the 
armature  be  allowed  to  move,  a  certain  e.m.f.  will  be  gen- 
erated at  the  brushes  due  to  the  fact  that  the  armature  conductors 
cut  the  field  magnetism  of  B.  This  generated  e.m.f.  will  at  each 
instant  be  proportional  to  the  product  of  the  field  magnetism 
and  the  speed,  and  will,  therefore,  be  in  time  phase  with  the 
magnetism  of  B.  Since  the  resistance  and  local  reactance  of 
the  armature  circuit  are  negligible,  any  unbalanced  e.m.f.  at 
the  brushes  would  cause  an  enormous  current  to  flow  through 
the  armature.  Such  current,  however,  would  produce  a  flux 
in  time  phase  with  itself  and  the  rate  of  change  of  the  flux  would 
generate  an  e.m.f.  opposing  the  effective  e.m.f.  which  causes 
current  to  flow,  the  final  result  being  that  there  flows  just  that 
amount  of  current,  the  magnetomotive  force  of  which  produces 


SINGLE-PHASE  COMMUTATOR  MOTORS. 


193 


a  value  of  flux  the  rate  of  change  of  which  through  the  armature 
generates  an  e.m.f.  equal  and  oppostie  to  that  due  to  the  speed. 
Now,  the  flux  thus  produced  alternates  through  the  winding 
on  A  and  generates  therein  an  e.m.f.  equal  to  that  similarly 
generated  at  the  brushes  and  in  time  phase  with  the  brush  e.m.f. 
Since  the  flux  at  B  is  in  time  phase  with  the  e.m.f.  across  the 
winding  on  B,  and  the  e.m.f.  counter-generated  in  the  winding 
on  A  is  in  time  phase  with  the  flux  in  B,  the  e.m.f.  across  the 
winding  B  is  in  time  quadrature  to  that  across  the  A  winding. 
The  vector  sum  of  the  e.m.fs.  at  A  and  at  B  must  be  equal  to 
the  constant  line  e.m.f.,  E. 


FIG.  102. — Vector  Diagram  of  Ideal  Repulsion  Motor. 

GRAPHICAL  DIAGRAM  OF  REPULSION  MOTOR. 

The  facts  just  stated  lead  to  the  very  simple  graphical  rep- 
resentation of  the  phenomena  of  an  ideal  repulsion  motor 
shown  in  Fig.  102,  where  A  C  represents  in  value  and  phase 
the  impressed  e.m.f.,  E;  the  line  A  B  equals  the  e.m.f.  across 
the  winding,  A,  at  a  certain  armature  speed,  while  B  C  is  the 
corresponding  e.m.f.  across  B  at  the  same  speed.  It  will  be 
noted  that  since  at  any  speed  the  e.m.fs.  A  B  and  B  C  must  be 
at  right  angles  and  have  a  vector  sum  equal  to  E,  the  locus  of 
the  point  B  is  a  true  circle  and  can  at  once  be  drawn  when  A  C 


194  ALTERNATING  CURRENT  MOTORS. 

is  located.  Since  the  e.m.f.  of  the  A  winding  is  proportional  to 
the  value  and  rate  of  change  of  the  flux  in  the  core  A,  and  such 
flux  is  proportional  to  the  current  in  the  coil,  the  current  in 
the  winding,  A,  is  proportional  to  the  e.m.f.,  A  B  (Fig.  102)  and 
in  time  quadrature  to  it,  as  shown  at  A  D.  A  little  considera- 
tion will  show  that  the  locus  of  D  is  a  true  circle  having  a  diam- 
eter A  F  equal  to  the  primary  current  at  standstill. 

It  has  been  stated  that  at  starting  the  primary  and  secondary 
currents  were  equal  in  value,  but  that  under  speed  conditions 
another  current  was  produced  in  the  secondary.  The  value  of 
this  additional  component  of  the  secondary  current,  which  must 
be  such  as  to  give  the  magnetism  in  A  (Fig.  101),  may  be  found 
as  follows:  The  magnetism  in  A  is  proportional  to  the  e.m.f. 
across  the  winding  of  those  poles.  Hence,  the  current  to  pro- 
duce such  magnetism  must  be  proportional  to  the  e.m.f.  and 
in  time  quadrature  to  it;  or,  if  in  Fig.  102,  B  C  is  the  e.m.f. 
just  referred  to,  C  G  is  the  component  of  secondary  current  to 
produce  the  corresponding  flux.  The  locus  of  G  is  a  true  circle 
with  a  diameter  equal  to  A  F  of  the  primary  current,  as  will  be 
apparent  from  what  has  been  stated  previously.  The  vector 
sum  of  the  components,  C  G,  of  the  secondary  current  and  C  H, 
equal  and  opposite  to  the  primary  current,  gives  the  true  sec- 
ondary current  both  in  value  and  phase  position. 

It  is  evident  that  at  a  certain  speed  the  e.m.fs.  represented  by 
the  lines  A  B  and  B  C  in  Fig.  102  will  be  equal  in  value.  It  is 
interesting  to  note  what  occurs  at  this  speed.  Since  the  e.m.fs. 
generated  are  equal  and  in  time  quadrature,  the  magnetic 
fluxes  must  similarly  have  equal  values  and  be  in  time  quad- 
rature. It  will  be  recalled  that  the  magnetic  condition  at  this 
speed  is  similar  in  all  respects  to  that  found  in  two-phase  in- 
duction motors;  that  is,  there  is  produced  a  true  uniform  ro- 
tating field  (if  a  continuous  core  be  used)  moving  at  "  synchron- 
ous speed."  At  other  armature  speeds  there  is  produced  sim- 
ilarly a  revolving  field  traveling  at  synchronous  speed,  but  the 
field  varies  in  intensity  from  instant  to  instant,  giving  what  is 
termed  an  "  elliptically  revolving  "  field,  one  axis  of  the  ellipse 
remaining  in  line  with  the  brushes  and  having  a  value  propor- 
tional to,  but  in  time  quadrature  with,  the  e.m.f.  of  the  winding 
of  B,  Fig.  101,  or  length  A  B  in  Fig.  102,  the  other  axis  being 
in  line  with  the  field  poles,  A,  and  proportional  to  the  e.m.f. 
across  the  winding  on  them,  B  C,  in  Fig.  102.  It  is  noteworthy 


SINGLE-PHASE  COMMUTATOR  MOTORS.  195 

that  such  an  elliptical  field  is  characteristic  of  the  magnetic 
condition  found  with  single-phase  induction  motors.  It  should 
be  noted,  however,  that  the  term  "  synchronous  speed  " 
refers  to  the  change  in  position  on  the  core  of  the  maximum 
magnetic  flux  existing  at  each  instant  and  has  no  direct  bearing 
upon  the  maximum  speed  which  the  rotor  may  attain,  which 
maximum  speed,  in  fact,  is  limited  by  conditions  other  than 
those  here  assumed  to  exist. 

CALCULATED  PERFORMANCE  OF  IDEAL  REPULSION  MOTOR. 

By  the  use  of  Fig.  102  the  complete  performance  of  an  ideal 
repulsion  motor  may  be  determined  if  values  be  assigned  to 
the  scales  chosen  for  the  current  and  e.m.f.  Table  I  records 
such  calculations  of  the  performance  of  a  certain  repulsion 
motor,  of  which  the  field  reactance  is  10  ohms  when  operated 
at  a  constant  impressed  e.m.f.  of  100  volts;  that  is,  in  Fig.  102, 
AC  =  100  V.,  A  F  =  10  A.  The  method  of  making  computa- 
tions is  indicated  at  the  head  of  each  column  of  the  table.  It 
will  be  observed  that  there  have  been  chosen  certain  values 
of  the  e.m.fs.  across  the  field  coils  on  the  poles  B  (Fig.  101) 
and  the  corresponding  values  of  the  e.m.fs.  across  the  trans- 
former coils  on  the  poles  A  have  been  calculated.  As  will  be 
seen,  the  work  of  determination  has  been  much  simplified  by 
assuming  e.m.f.  values  corresponding  to  the  product  of  the 
impressed  e.m.f.  and  the  sine  and  cosine  values  of  angles  at  five- 
degree  intervals,  so  that  almost  all  values  recorded  in  Table  I 
have  been  taken  directly  from  trigonometric  tables. 

The  speed  is  found  as  follows:  With  an  equal  number  of  turns 
in  the  windings  on  A  and  B  of  Fig.  101,  when  the  e.m.fs.  across 
these  windings  are  equal,  the  speed  is  synchronous,  and  this 
speed  is  given  the  value  1.  Now,  at  other  speeds,  the  e.m.f. 
across  the  A  winding  will  be  proportional  to  the  product  of  the 
speed  and  the  flux  in  B;  that  is,  the  e.m.f.  in  the  winding,  B, 
as  shown  previously.  It  will  be  apparent,  therefore,  that  the 
speed  is  the  ratio  of  the  e.m.f.  across  the  coils  on  A  to  that 
across  the  coils  on  B  of  Fig.  101,  or  to  the  ratio  of  the  lengths 
B  C  to  A  B  in  Fig.  103.  It  will  be  noted  that  this  ratio  is  the 
cotangent  of  the  angle  A  C  B  arbitrarily  chosen  at  five-degree 
intervals,  so  that  the  speed  may  be  taken  at  once  from  trigo- 
nometric tables.  The  torque  is  expressed  in  synchronous 
watts,  as  the  ratio  of  output  to  speed. 


196 


ALTERNATING  CURRENT  MOTORS 

O(_  5  &  op  3>  ^  *•*  t*  CQ  op  GO  **  o  9  t*»  «H.IO  o  S 

"  <L>    £3    On 

05  CO  0-~, 

L    o* 


b^ 


OOOOOf-H»HrHrHr-<^-lC^NW<NW<N(N 


O  IQ  tt  ^0  9  tQ  ^tt  t«  *V  QD  »«t  ^0  O  O  00  tt  ^0 


OOOOOOOOOOOOOOOOi-HN 


05       0-ft, 


a 


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r'- 

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W'o 


88§S8§8SS§§S§§§§88 

ooooooooooooooooo'o' 


OOOOOOOOOOOOOOOONO 


Or-i(MCO-*iot^OOO'-H^?3^^K?O^ 


OO»OOOOOO»OOiOOOOOOiCO 

O(NCOOOOCOO«OC»'— iO<NOCOt^C5OO<N 
Ot>-COOO(M(NOfOINt>.OC5CDCDO>»O'*CC 


o 

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O    C^l 

8S 


SINGLE-PHASE  COMMUTATOR  MOTORS. 


197 


Columns  6  to  11  of  Table  I  have  been  calculated  on  the  as- 
sumption that  the  brush  angle  a  of  Fig.  100  was  45°,  or  what  is  the 
same,  that  the  number  of  turns  on  field  poles  B  was  equal  to 
that  on  the  transformer  poles,  A,  Fig.  101,  and  these  calculations 
are  graphically  represented  in  Fig.  103.  In  Fig.  104  are  given 
the  characteristics  of  an  ideal  repulsion  motor  with  the  brush 
angle  a  of  Fig.  100  having  a  value  such  that  its  tangent  is  0.25, 
and  the  calculations  recorded  in  columns  6'  to  11"  are  based 
on  this  assumption.  The  conditions  here  assumed  are  equiva- 
lent to  what  would  be  obtained  if  the  number  of  turns  on  the 
field  poles,  B,  were  0.25  of  the  turns  on  the  transformer  poles, 
A,  Fig.  101.  It  is  further  assumed  that  the  coils  on  poles  B 


0    .1    3,    .3     .4     .5 


Speed 


FIG.  103. — Characteristics  of  Ideal  Repulsion  Motor. 

are  the  same  in  number  as  previously;  that  is,  that  the  field 
reactance  is  10  ohms  as  before,  and  that  the  turns  on  both  the 
armature  and  transformer  poles  have  been  increased  to  four 
times  their  first  value. 

The  method  of  calculation  is  quite  the  same  as  before,  but 
perhaps  a  word  is  needed  as  to  the  calculation  of  the  speed  and 
secondary  current.  At  synchronous  speed,  the  magnetism  in 
the  field  poles,  B,  is  equal  to  that  in  the  transformer  poles,  A, 
as  noted  above.  Due  to  the  increased  number  of  turns  on  the 
transformer  poles,  at  synchronous  speed,  the  e.m.f.  across  the 
coils  on  A  will  be  four  times  that  across  the  coils  on  B,  Fig.  101. 
A  consideration  of  this  fact  leads  to  the  conclusion  that  the  speed 
is  all  times  equal  to  one-fourth  of  the  ratio  of  B  C  to  A  B  in 


198 


ALTERNATING  CURRENT  MOTORS. 


Fig.  102;  that  is,  the  speed  is  equal  to  one-fourth  of  the  co- 
tangent of  the  angle  6,  arbitrarily  assumed,  and  hence  may 
quite  readily  be  computed. 

The  component  of  secondary  current  to  counterbalance  the 
primary  current  is  equal  and  opposite  to  the  primary  current, 
since  the  armature  turns  are  equal  in  number  to  the  trans- 
former turns  on  A ;  but  the  current  to  produce  the  magnetism 
in  the  transformer  poles  under  speed  conditions  will  at  all  times 
be  one-fourth  the  value  required  to  produce  the  same  mag- 
netism in  the  field  poles,  as  will  be  seen  from  column  10 '  of  the 
table. 

A  comparison  of  the  curves  of  Fig.  103  with  those  of  Fig.  104 


FIG.  104. — Characteristics  of  Ideal  Repulsion  Motor. 

will  reveal  the  effect  of  shifting  the  brushes  from  the  45°  position 
farther  toward  the  axial  line  of  the  transformer  poles.  It  is 
essential  for  good  performance  that  the  angle  of  brush  shift 
from  the  transformer  position  be  quite  small,  usually  from  12 
to  16°,  depending  upon  the  constructive  constants  of  the  machine. 
It  must  be  very  carefully  noted  that  the  curves  here  given 
are  for  an  ideal  repulsion  motor,  all  resistance,  local  inductance 
and  short-circuiting  effects  having  been  neglected,  and  that 
such  curves  cannot  be  realized  in  practice.  It  is  worthy  of 
note,  however,  that  upon  the  characteristics  here  shown  are 
based  the  discussions  of  the  properties  of  the  repulsion  motor 
which  have  occupied  so  much  space  in  the  technical  papers. 


CHAPTER  XIV. 

MOTORS  OF  THE  REPULSION  TYPE  TREATED  BOTH 
GRAPHICALLY  AND  ALGEBRAICALLY. 

ELECTROMOTIVE  FORCES  PRODUCED  IN  AN  ALTERNATING  FIELD. 

In  dealing  with  the  phenomena  connected  with  the  operation 
of  alternating  current  motors  of  the  commutator  type,  it  must 
be  constantly  borne  in  mind  that  the  machine  possesses  simul- 
taneously the  electrical  characteristics  of  both  a  direct  current 
motor  and  a  stationary  alternating  current  transformer.  The 
statement  just -made  must  not  be  confused  with  a  somewhat 
similar  one  which  is  applicable  to  polyphase  induction  motors, 
since  only  with  regard  to  its  mechanical  characteristics  does  an 
induction  motor  resemble  a  shunt-wound  direct  current  ma- 
chine, its  electrical  characteristics  being  equivalent  in  all  re- 
spects to  those  of  a  stationary  transformer. 

Before  discussing  the  performance  of  repulsion  motors,  it  is 
well  to  investigate  a  few  of  the  properties  common  to  all  com- 
mutator type,  alternating  current  machines.  It  will  be  recalled 
that  when  the  current  flows  through  the  armature  of  a  direct 
current  machine,  magnetism  is  produced  by  the  ampere  turns  of 
the  armature  current,  such  magnetism  tending  to  distort  the 
flux  from  the  field  poles.  In  the  familiar  representation  of  the 
magnetic  circuit  of  machines, — the  two  pole  model, — the  arma- 
ture magnetism  is  at  right  angles  to  the  field  magnetism,  the 
armature  current  producing  magnetic  poles  in  line  with  the 
brushes.  The  amount  of  this  magnetism  depends  directly  on 
the  value  of  the  armature  current  and  the  permeability  of  the 
magnetic  path.  When  alternating  current  is  used,  the  change 
of  the  magnetism  with  the  periodic  change  in  the  current  pro- 
duces an  alternating  e.m.f.  which  being  proportional  to  the  rate 
of  change  of  the  magnetism  will  be  in  time-quadrature  to  the 
current.  The  armature  winding  thus  acts  in  all  respects  sim- 
ilarly to  an  induction  coil. 

It  is  not  essential  that  the  current  to  produce  the  alternating 

199 


200 


ALTERNATING  CURRENT  MOTORS. 


flux  flow  through  the  armature  coils  in  order  that  the  alter- 
nating e.m.f.  be  developed  at  the  commutator.  Under  whatso- 
ever conditions  the  armature  conductors  be  subject  to  changing 
flux  a  corresponding  e.m.f.  will  be  generated,  in  mechanical  line 
with  the  flux  and  in  time-quadrature  to  it.  Referring  to  Fig.  105 
which  represents  a  direct  current  armature  situated  in  an  alter- 
nating field,  having  two  pair  of  brushes,  one  in  mechanical  line 
with  the  alternating  flux  and  one  in  mechanical  quadrature 
thereto.  When  the  armature  is  stationary  an  e.m.f.  will  be 
generated  at  the  brushes  A  and  A  due  to  the  transformer  action 


Transformer 
E.M.F. 


Speed  E.M.F. 
FIG.  105. — Electromotive  Forces  Produced  in  an  Alternating  Field. 

of  the  flux,  but  no  measurable  e.m.f.  will  exist  between  B  and 
B.  As  seen  above,  this  e.m.f.  is  in  time-quadrature  with  the 
field  (transformer)  flux  'and  as  will  be  seen  later,  its  value  is  un- 
altered by  any  motion  of  the  armature.  At  any  speed  of  the 
armature,  there  will  be  generated  at  the  brushes  B  and  B  an 
e.m.f.  proportional  to  the  speed  and  to  the  field  magnetism  and 
in  time-phase  with  the  magnetism.  At  a  certain  speed  this 
"  dynamo  "  e.m.f.  will  be  equal  in  effective  value  to  the  "  trans- 
former "  e.m.f.  at  A  and  A,  though  it  will  be  in  time-quadrature 
to  it.  This  critical  speed  will  hereafter  be  referred  to  as  the 


REPULSION  MOTORS.  201 

"  synchronous  "  speed,  and  with  the  two-pole  model  shown 
in  Fig.  105  it  is  characterized  by  the  fact  that  in  whatsoever 
position  on  the  armature  a  pair  of  brushes  be  placed  across  a 
diameter,  the  e.m.f.  between  the  two  brushes  will  be  the  same 
and  will  have  a  relative  time-phase  position  corresponding  to 
the  mechanical  position  of  the  brushes  on  the  commutator. 

A  little  consideration  will  show  that  the  individual  coils  in 
which  the  maximum  e.m.f.  is  generated  by  transformer  action 
are  situated  upon  the  armature  core  under  brush  B  or  B,  al- 
though the  difference  of  potential  between  the  brushes  B  and  B 
is  at  all  times  of  zero  value  as  concerns  the  transformer  action. 
A  similar  study  leads  to  the  conclusion  that  the  e.m.f.  generated 
by  dynamo  speed  action  appears  as  a  maximum  for  a  single  coil 
when  the  coil  is  under  brush  A  or  A.  Assuming  as  zero  posi- 
tion, the  place  under  brush  A  and  that  at  synchronous  speed 
the  e.m.f.  generated  in  a  coil  at  this  position  is  e.  Then  the 
e.m.f.  in  a  coil  at  b  will  equal  e  also.  A  coil  a  degrees  from  this 
position  will  have  generated  in  it  a  speed  e.m.f.  of  e  cos  a  and  a 
transformer  e.m.f.  of  e  cos  (a  ±  90)  =  ±  e  sin  a.  Since  these 
two  component  e.m.fs.  are  in  time  quadrature  the  resultant  will 
be  V  =  \/(e  cos  a)2+  (±  e  sin  a)2  =  e  and  is  the  same  for  all 
values  of  a.  The  time-phase  position  of  the  resultant,  however, 
will  vary  directly  with  a  or  with  the  mechanical  position  of  the 
coil.  From  these  facts  it  is  seen  that  at  synchronous  speed  the 
effective  value  of  the  e.m.f.  generated  per  coil  at  all  positions 
is  the  same  and  that  there  is  no  neutral  e.m.f.  position  on  the 
commutator. 

THE  SIMPLE  REPULSION  MOTOR. 

In  a  repulsion  motor  as  commercially  constructed,  the  sec- 
ondary consists  of  a  direct  current  armature  upon  the  commu- 
tator of  which  brushes  are  placed  in  positions  180  electrical  de- 
grees apart  and  directly  short  circuited  upon  themselves,  as 
shown  in  the  two-pole  model  of  Fig.  106.  The  stationary  pri- 
mary member  consists  of  a  ring  core  containing  slots  more  or 
less  uniformly  spaced  around  the  air-gap.  In  these  slots  are 
placed  coils  so  connected  that  when  current  flows  in  them  defi- 
nite magnetic  poles  will  be  produced  upon  the  field  core.  The 
brushes  on  the  commutator  are  given  a  location  some  15  degrees 
from  the  line  of  polarization  of  the  primary  magnetism,  or 


202 


ALTERNATING  CURRENT  MOTORS. 


more  properly  expressed,  the  brushes  are  placed  about  15  de- 
grees from  the  true  transformer  position.  That  component  of 
the  magnetism  which  is  in  line  with  the  brushes  produces  cur- 
rent in  the  secondary  by  transformer  action,  and  this  current 
gives  a  torque  to  the  rotor  due  to  the  presence  of  the  other  com- 
ponent of  magnetism  in  mechanical  quadrature  to  the  secondary 
current. 

It  is  possible  to  make  certain  assumptions  as  to  the  relative 
values  of  the  magnetism  in  mechanical  line  with,  and  in  me- 
chanical quadrature  to  the  brush  line  and  thus  to  derive  the 


4 
E 

pressed 
.M.F. 

_i  

ET   < 

Transformer, 
E.M.F.    ( 

Field          \        ' 

liausioruier 

KM.F. 

^-  **x 

Field 


FIG.  106. — Two-pole  Model  of  Ideal  Repulsion  Motor. 

fundamental  equations  of  the  machine.  It  is  believed,  how- 
ever, that  the  facts  can  be  more  clearly  presented  and  the  treat- 
ment simplified  without  sacrifice  of  accuracy  if  the  assumption 
be  made  that  the  primary  coil  is  wound  in  two  parts,  one  in  me- 
chanical line  and  the  other  in  mechanical  quadrature  with  the 
axial  brush  position  as  shown  in  Fig.  106.  It  will  be  noted  that 
the  two  fields  produced  by  the  sections  of  the  primary  coil,  if 
there  were  no  disturbing  influence  present,  would  have  a  result- 
ant position  relative  to  the  brush  line  depending  upon  the  ratio 
of  the  strengths  of  the  two  magnetisms.  The  angle  which  the 
resultant  field  would  assume  can  be  represented  by  /?  having  a 


REPULSION  MOTORS.  203 

value  such  that  cotan  /?  =  ^£  where  </.>,  is  the  flux  through  trans- 
former coil  and  <£/  is  flux  through  field  coil.  If  n  be  the  ratio  of 
turns  on  the  transformer  poles  to  those  on  the  field  poles,  then 
for  any  value  of  current  in  these  coils  (no  secondary  current) 

<ht 

~-  =  n  or  n  =  cotan  f)  (1) 

It  is  understood  that  in  Fig.  106,  the  core  material  is  consid- 
ered to  be  continuous  and  that  in  the  two-pole  model  represented 
both  field  poles  and  both  transformer  poles  are  supposed  to  be 
properly  wound. 

In  Fig.  106,  let  it  be  assumed  that  the  machine  is  stationary 
and  that  a  certain  e.m.f.,  E,  is  impressed  upon  the  primary  cir- 
cuits, the  secondary  being  on  short  circuit.  The  flux  which  the 
primary  current  tends  to  produce  in  the  transformer  pole  pro- 
duces by  its  rate  of  change  an  e.m.f.  in  the  secondary,  and  this 
e.m.f.  causes  opposing  current  to  flow  in  the  closed  secondary 
circuit.  If  the  transformer  action  is  perfect  and  the  trans- 
former coil  and  armature  circuits  are  without  resistance  and 
local  leakage  reactance,  then  the  magnetomotive  force  of  the 
armature  current  equals  that  of  the  current  in  the  transformer 
coil,  and  the  resultant  impedance  effect  of  the  two  circuits  is 
of  zero  value,  so  that  the  full  primary  e.m.f.,  E,  is  impressed 
upon  the  field  coil;  that  is  to  say,  with  armature  stationary 
Et  =  O,  and  Ef  =  E. 

EFFECT  OF  SPEED  ON  THE  STATOR  ELECTROMOTIVE  FORCES. 

It  remains  now  to  investigate  the  effect  of  speed  on  the 
electromotive  forces  of  the  transformer  and  field  coils.  Assume 
a  certain  flux  0  in  the  field  coil.  At  sneed  5  the  armature  con- 
ductors will  cut  this  flux  and  at  each  instant  there  will  be  gen- 
erated an  e.m.f.  therein  proportional  to  S  <f>f,  and  therefore,  in 
time-phase  with  the  flux.  This  e.m.f.  would  tend  to  cause  cur- 
rent to  flow  in  the  closed  armature  circuit,  which  current  would 
produce  magnetism  in  line  with  the  brushes,  and,  since  the 
armature  circuit  has  zero  impedance,  (assumed)  the  flux  so  pro- 
duced will  be  of  a  value  such  that  its  rate  of  change  through  the 
armature  coils  just  equals  the  e.m.f.  generated  therein  by  speed 
action.  At  synchronous  speed,  the  secondary  being  closed,  the 


204  ALTERNATING  CURRENT  MOTORS. 

flux  in  line  with  the  brushes  must  equal  that  in  line  with  the 
field  poles,  since  the  e.m.f.  generated  by  the  rate  of  change  of 
the  flux  in  the  direction  of  the  brushes  must  equal  that  gen- 
erated at  the  brushes  due  to  cutting  the  field  magnetism,  and  at 
a  speed  which  has  been  termed  synchronous  these  two  fluxes 
are  equal,  as  previously  discussed.  At  this  speed  the  two  fluxes 
are  equal  but  they  are  in  time-quadrature  one  to  the  other. 
At  other  speeds  the  two  fluxes  retain  the  quadrature  time-phase 
position,  but  the  ratio  of  the  effective  values  of  the  two  fluxes 
varies  directly  with  the  speed. 

FUNDAMENTAL  EQUATIONS  OF  THE  REPULSION  MOTOR. 

Giving  to  synchronous  speed  a  value  of  unity,  at  any  speed, 
5,  the  transformer  flux  may  be  expressed  by  the  equation 

<f>t  =  S<t>f  (2) 

effective  values  being  used  throughout.  Letting  (f>  be  the  max- 
imum values  of  the  field  flux  and  reckoning  time  in  electrical  de- 
grees from  the  instant  when  the  field  flux  is  maximum,  at  any 
time  #,  the  instantaneous  field  flux  is 

$f  =  <l>  cos  ^  (3) 

and  the  transformer  flux  is 

(j)t  =  S  <f>  sin  H  (4) 

These  are  the  fundamental  magnetic  equations  of  the  ideal 
repulsion  motor. 

If  at  a  certain  speed  5,  the  effective  value  of  e.m.f.  across  the 
field  coil  be  F,  requiring  an  effective  flux  of  <£/,  then  across  the 
transformer  coil  there  will  be  an  effective  e.m.f.  of 

T  =  n  S  F  (5) 

due  to  the  flux  S  (j)f.  Since  the  fluxes  are  in  time-quadrature, 
the  e.m.fs.  are  likewise  in  time  quadrature,  so  that  the  impressed 
e.m.f.  E  must  have  a  value  such  that 

E  =  x/FH^  (6) 

This  is  the  fundamental  electromotive  force  equation  of  the 
repulsion  motor. 

The  current  which  flows  through  the  field  coil  is 

'-  "       (7) 


REPULSION  MOTORS.  205 

where  X  is  the  inductive  reactance  of  the  field  coil.  Equation 
(7)  gives  the  value  of  the  primary  circuit  current  and  is  the 
fundamental  primary  current  equation. 

The  secondary  armature  current  in  general  consists  of  two 
components,  that  equal  in  magnetomotive  force  and  opposite  in 
phase  to  the  primary  transformer  current,  and  that  necessary 
to  produce  the  flux  in  line  with  the  brushes.  With  a  ratio  of 
effective  armature  turns  to  field  turns  of  a,  the  opposing  trans- 
former current  is 

-.      •  '  fc-*T  (8) 

and  the  current  which  produces  the  transformer  poles  is 

If  =  IT  (9) 

These  component  currents  are  in  time-quadrature,  so  that  the 
resultant  secondary  current  is 

fa  =  Vl?  +  If  (10) 

This  is  the  fundamental  equation  for  the  secondary  current. 
Combining  (8),  (9)  and  (10) 

r        I    , 

fa  =—Vn2  +  S2  (11) 

a 

It  has  been  seen  that  the  e.m.f.  T  is  in  time-quadrature  to  the 
field  circuit  e.m.f.,  F.  Now  the  current  is  in  time-quadrature 
with  F,  and  hence,  is  in  time-phase  with  T.  Therefore,  of  the 
total  primary  e.m.f.  E,  the  part  T  is  in  phase  with  the  current, 
from  which  fact  it  is  seen  that  the  power  factor  is 

cos  0  =  J  (12) 

Power, 

P  =  E  I  cos  6  =  E^£     =  /  T  (13) 

Torque, 

P        IT       ISnF 


206  ALTERNATING  CURRENT  MOTORS. 

D  =  InF  =  InXI  =  PnX  (14) 

E2  =  F2  +  P  =  F2  (1 +52  n2)  (15) 

F  =  vpW    III;  (16) 

7  =  y./f.  „    ,  (l^7) 


~  aXVT+S^* 
when  n  =  1,  that  is  at  /9  =  45°  see  (1) 

£ 

/„  =  — ^>  and  is  constant  at  all  speeds. 
<z  A 

when  5=1,  that  is  at  synchronism  for  any  value  of  n, 

I«  =  — ^.  which  is  seen  to  be  equal  to  the  primary 

current  at  starting  (when  a  =  1) 
when  5=1  the  secondary  current 

/      , ; 

la  ='-  —  vV+1 

and  leads  the  primary  current  by  angle  cot  an-1  n  =  /?  or  angle 
of  brush  shift.     See  equation  (1). 

VECTOR  DIAGRAM  OF  IDEAL  REPULSION  MOTOR. 

The  above  equations  can  be  expressed  graphically  by  a  simple 
diagram  as  shown  in  Fig.  107.  The  diagram  is  constructed  as 
follows:  O  E  is  the  constant  line  e.m.f.  O  A  at  rt.  angles  to 
O  E  is  the  line  current  at  starting,  O  B  A  is  a  semicircle,  O  F 
in  phase  opposition  to  O  A  is  the  secondary  current  at  starting. 
ODF  is  a  semicircle.  OG,  in  phase  with  O  A,  is  the  sec- 
ondary current  at  infinite  speed.  0  H  G  is  a  semicircle.  It  will 
be  noted  that  the  ratio  0  A  to  0  G  is  n  a :  1  and  ratio  of  0  A  to 
OF  is  a\n. 

Distances  measured  from  P  in  the  direction  of  T  represent 
speed. 

The  characteristics  of  the  machine  may  be  found  at  once  from 


REPULSION  MOTORS. 


207 


Fig.  107.  Assuming  any  speed  as  PS,  draw  OS  intersecting 
the  circle  O  B  A  at  B.  From  point  G  draw  line  G  K  parallel 
to  OS.  Join  O  and  K. 

0  K  is  secondary  current ; 

O  B  is  primary  current ; 

E  O  S  is  primary  angle  of  lag ; 

B  C  is  power  component  of  primary  current ; 


FIG.  107. — Vector  Diagram  of  Ideal  Repulsion  Motor. 

B  C  is  power  (to  proper  scale) ; 
O  C  is  torque  (to  proper  scale) ; 
D  0  K  is  angle  of  lead  of  secondary  current. 
At  synchronous  speed   (S  =  1)   cotan  0  =  n,  hence  scale  of 
speed  can  readily  be  located. 
O  D  =  It,  see  equation  (8) 
0  H  =  If.  see  equation  (9). 


208  ALTERNATING  CURRENT  MOTORS. 

The  proof  of  the  construction  of  diagram  of  Fig.  107  is  as 
follows: 

f-  Eq.  (11) 

E2  =  T*+F2  Eq.  (6) 

T  =  5  n  F  Eq.  (5) 

(19) 


E  =  ~  x/l+S2n2  (20) 


cos  0  =        =     x  (21) 

E        \/l+S2w2 

Power  component  of  primary  current    : 

.'  /,-/cbS«-f  (22) 


Quadrature  component  of  primary  current 

E  1 


Iq  =  I  sin  0  = 


\/l+52n? 


7(7  ~  X(l+52n2) 


cotan  6  =  S  n  (25) 

The  cotangent  of  the  angle  of  lag  is  directly  proportional  to 
the  speed,  the  proportionality  constant  being  the  ratio  of  trans- 
former to  field  turns. 

_.       IE  cos  0  E2  n  ~  T 

>    =  nEI<  <26) 


REPULSION  MOTORS.  209 

Torque  is  proportional  to  quadrature  component  of  the  pri- 
mary current  (for  given  e.m.f.)  the  proportionality  constant 
being  the  ratio  of  transformer  to  field  turns. 


Torque  varies  as  the  square  of  the  primary  current  and  in  this 
respect  is  independent  of  the  speed  or  the  e.m.f. 

A  comparison  of  equations  (26)  and  (27)  reveals  an  interest- 
ing property  of  a  circle.  In  Fig.  107,  assuming  the  diameter 
A  0  to  be  unity,  0  C  at  all  values  of  angle  6  equals  the  square 
of  OB. 

From  equation  (27)  it  is  seen  that  the  torque  is  at  all  times 
positive,  even  when  S  is  negative.  Hence  the  machine  acts  as 
generator  at  negative  speed.  For  the  determination  of  the 
generator  characteristics  it  is  necessary  to  construct  the  semi- 
circle omitted  in  each  case  in  Fig.  107. 

It  is  interesting  to  observe  that  the  construction  of  the  dia- 
gram of  Fig.  107  can  be  completed  at  once  when  points  F,  0,  G 
and  A  and  E  are  located.  Thus  the  complete  performance  of 
the  ideal  repulsion  motor  can  be  determined  when  E,  X,  n  and 
a  are  known.  In  the  construction  for  ascertaining  the  value  of 
the  secondary  current,  it  will  be  seen  that  O  K  is  equal  to  the 
vector  sum  of  O  D  and  0  H,  giving  the  vector  O  K.  From  the 
properties  of  vector  co-ordinates  it  will  be  noted  that  the  point 
K  is  located  on  the  semicircle  F  K  G  whose  center  lies  in  the 
line  FOG.  Therefore  if  G  and  F  be  located,  the  inner  circles 
F  D  0  and  0  H  G  need  not  be  drawn,  since  the  point  K  can  be 
found  as  the  intersection  of  the  line  drawn  parallel  to  0  B  from 
G  with  the  circular  arc  F  K  G. 

CORRECTIONS  FOR  RESISTANCE  AND  LOCAL  LEAKAGE  REACTANCE. 

It  is  to  be  carefully  noted  that  the  above  discussion  refers  to 
ideal  conditions  which  can  never  be  realized.  The  circuits  have 
been  considered  free  from  resistance  and  leakage  reactance  while 
all  iron  losses,  friction,  and  brush  short  circuiting  effects  have 
been  neglected.  The  resistance  and  leakage  reactance  effects 
can  quite  easily  be  taken  into  account,  but  the  remaining  dis- 
turbing influences  are  subject  to  considerable  error  in  approxi- 
mating their  values,  due  primarily  to  the  difficulty  in  assigning 


210  ALTERNATING  CURRENT  MOTORS. 

to  iron  any  constant  in  connection  with  its  magnetic  phenomena. 
It  is  to  be  regretted  that  the  so-called  complete  equations  for 
expressing  the  characteristics  of  this  type  of  machinery  with 
almost  no  exception  neglect  these  disturbing  influences,  and  yet 
these  same  equations  are  given  forth  by  the  various  writers  as 
though  they  represented  the  true  conditions  of  operation. 
In  the  ideal  motor  the  apparent  impedance  is 

S*n*  (28) 


apparent  resistance  is 

R  =  ZcosO  =  XSn  (29) 


smcc 


hence 


cos#  =  — ;  T  =  S  nF\  and  E  = 


„         T  S  n 

JS    =   7^ —  \/  1  4-  S2  n2 ',    COS  U   =    - 

s  H  \   i-t-3   n 


apparent  reactance  is 

X  =  Zsintf  =  X  (30) 


since 


S2 


Let  Rf   =  resistance  of  field  coil 

RI    =  resistance  of  transformer  coil 
Ra  =  resistance  of  armature  coil 
Xa  =  reactance  of  armature  coil 
Xt  =  reactance  of  transformer  coil 
Xj  =  reactance  of  field  coil 
then  copper  loss  of  motor  circuits  will  be 

P  (Rj  +  Rt)+Iu2Ra  (31) 


X 


(17) 


REPULSION  MOTORS.  211 

hence 

In  -  /X/nJ  +  52  (32) 

and  copper  loss  will  be 

li  +  £,)  +  ^ — — J  Ra  J  ==  72  tfw  (33) 

where  Rm  is  the  effective  equivalent  value  of  the  motor  circuit 
resistance,  that  is 


j-)  E>_i_E>iflZ?  /°_1\ 

Km  =  Kf  +  Kt-r(        1  Ka  {&*) 


Similarly  it  may  be  shown  that  the  effective  equivalent  value 
of  the  leakage  reactance  of  the  motor  circuits  is 

t«  (35) 

If  these  values  be  added  to  the  apparent  resistance  and  react- 
ance of  the  ideal  motor  the  corresponding  effects  will  be  repre- 
sented in  the  resultant  equations  thus 


Ra  (36) 

and 

X  =  X  +  Xf  +  Xt      --    Xa  (37) 


V~R*~+~X'2  from  (36)  and  (37)  (38) 


R  E 

eosfiL—  ;  I  =  ~  (39) 


Input  =  E  I  cos  0  (40) 

output  =  EIcosO  —  rR,n=sP  (41) 

p 

torque  =  —  =  D,  etc.  (42) 


212  ALTERNATING  CURRENT  MOTORS. 

BRUSH  SHORT-CIRCUITING  EFFECT. 

It  will  be  noted  that  the  short  circuiting  by  the  brush  of  a 
coil  in  which  an  active  e.m.f.  is  generated  has  thus  far  not  been 
considered.  Referring  to  Fig.  106,  it  will  be  seen  that  at  any 
speed  S  there  will  be  generated  in  the  coil  under  the  brush  by 
dynamo  speed  action  an  e.m.f. 

Es  =  K  fa  S  (43) 

where  K  is  constant.  This  e.m.f.  is  in  time-phase  with  the 
flux  </>,.  In  this  coil  there  will  also  be  generated  an  e.m.f.  by 
the  transformer  action  of  the  field  flux,  such  that, 

Ef  =  K  fa  (44) 

This  e.m.f  is  in  time-quadrature  to  <£/.  Since  <£/  and  <£/  are 
in  time  quadrature  the  component  e.m.fs.  acting  in  the  coil 
under  the  brush  are  in  time-phase  (opposition)  so  that  the  re- 
sultant e.m.f.  is 

Eb  =  Ef-E5  =  K  (fa  -  S  fa)  (45) 

Eb  =  K  fa  (1  -  S2)  Eq.  (2)  (46) 

Since  for  constant  frequency  of  supply  current,  F  is  propor- 
tional to  fa  we  may  write  fa  =  CF,  C  being  a  constant  depend- 
ing on  the  number  of  field  turns. 

CE  (47) 


hence 

KCE  (1-S2) 


which  becomes  zero  at  ±  S  =  1,  that  is  at  synchronizing  when 
operated  as  either  a  motor  or  a  generator.  Above  synchronism 
EI  increases  rapidly  with  increase  of  speed. 

The  friction  loss  can  best  be  taken  into  account  by  consider- 
ing the  friction  torque  as  constant  (  =  d)  and  subtracting  this 
value  from  the  delivered  electrical  torque  so  that  the  active 
mechanical  torque  becomes, 

Torque  =  D-d  (49) 

While  the  effect  of  the  iron  loss  is  relatively  small  as  concerns 
the  electrical  characteristics  of  the  machine  it  is  obviously  in- 


REPULSION  MOTORS.  213 

correct  to  neglect  it  when  determining  the  efficiency.  For  pur- 
pose of  analysis  it  is  convenient  to  divide  the  core  material  into 
three  parts,  the  armature  the  field  and  the  transformer  portions. 
Since  the  frequency  of  the  reversal  of  the  flux  in  both  the  trans- 
former and  the  field  portions  is  constant  the  losses  therein  will 
depend  only  upon  the  flux.  Thus  considering  hysteresis  only, 
the  transformer  iron  loss  is 

Ht  =  L  <}>™  (50) 

where  L  is  a  constant  depending  upon  the  mass  of  the  core 
material.  Similarly  the  field  iron  loss  is 

Hf  =  M<l>?'*  (51) 

M  being  a  constant 

Ht  +  Hf  =  fr™  (M  +  SL)  Eq.  (2)  (52) 

Since  both  the  field  and  the  transformer  fluxes  pass  through 
the  armature  core  and  these  two  fluxes  are  of  the  same  frequency 
but  displaced  in  quadrature  both  in  mechanical  position  and  in 
time-phase  relation,  the  resultant  is  an  elliptical  field  revolving 
always  at  synchronous  speed,  having  one  axis  in  line  with  the 
transformer  and  the  other  in  line  with  the  field,  the  values  being 
\/2~<£<  and  \/12<t>f  respectively.  The  value  of  the  two  axes  may 
be  written  thus 

\/2~S  (j)f  and  x/2~$f 

At  synchronous  speed  of  the  armature  the  two  become  equal 
and  since  no  portion  of  the  iron  is  then  subjected  to  reversal 
of  magnetism  the  iron  loss  of  the  armature  core  is  of  zero  value. 
At  other  speeds,  while  the  revolving  elliptical  field  yet  travels 
synchronously,  the  armature  does  not  travel  at  the  same  speed, 
so  that  certain  sections  of  the  armature  core  are  subjected  to 
fluctuations  of  magnetism  while  others  are  subjected  to  com- 
plete reversals,  the  sections  continually  being  interchanged. 
It  is  due  to  this  fact  that  no  correct  equation  can  be  formed  to 
represent  the  core  loss  of  the  armature  at  all  speeds,  since  the 
behavior  of  iron  when  subjected  to  fluctuating  magnetism  cannot 
be  reduced  to  a  mathematical  expression. 

OBSERVED  PERFORMANCE  OF  REPULSION  MOTOR. 

Fig.  108  shows  the  observed  characteristics  of  a  certain  four- 
pole  repulsion  motor  when  operated  at  22J  cycles,  the  syn- 
chronous speed  being  675  r.p.m.  It  will  be  noted  that  up  to 


214 


ALTERNATING  CURRENT  MOTORS. 


either  positive  or  negative  synchronism  the  apparent  reactance 
is  practically  constant  and  the  resistance  varies  directly  with 
the  speed,  but  that  beyond  synchronism  the  reactance  tends 
to  increase  and  the  resistance  is  no  longer  proportional  to  the 
speed,  the  power  factor  tending  to  decrease.  The  detrimental 
effects  above  synchronism  may  be  attributed  largely  to  the  e.m.f . 
generated  in  the  coil  short  circuited  by  the  brush,  as  indicated 
in  equation  (48). 

COMPENSATED  REPULSION  MOTOR. 

A  type  of  motor  closely  related  to  the  repulsion  machine  in 
the   performance   of  its  magnetic   circuits  is  the   compensated 


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Speed  in  100  R.P.  M, 

FIG.  108.— Test  of  Repulsion  Motor. 

repulsion  motor  shown  in  Fig.  109.  Its  electrical  circuits  seem 
to  be  those  of  a  series  machine  with  the  addition  of  a  second 
set  of  brushes,  A  A,  placed  in  mechanical  line  with  the  field  coil 
and  short-circuited  upon  themselves.  The  transformer  action 
of  this  closed  circuit  is  such  that  the  real  power  which  the  motor 
receives  is  transmitted  to  the  armature  through  this  set  of 
brushes,  while  the  remaining  set,  B  B,  which  in  the  plain  series 
motor  receives  the  full  electrical  power  of  the  machine,  here 
serves  to  supply  only  the  wattless  component  of  the  apparent 
power.  This  complete  change  in  the  inherent  characteristics 
of  the  series  machine  by  the  mere  addition  of  two  brushes  ren- 
ders the  study  of  this  type  of  motor  especially  interesting. 


REPULSION  MOTORS. 


215 


For  purpose  of  analysis,  assume  an  ideal  motor  without  re- 
sistance or  local  leakage  reactance  and  consider  first  the  con- 
ditions when  the  armature  is  at  rest.  When  a  certain  e.m.f.,  E, 
is  impressed  upon  the  motor  terminals,  the  counter  magnetizing 
effect  of  the  current  in  the  brush  circuits,  A  A,  s  such  that  the 
e.m.f.  across  the  transformer  coil  is  of  zero  value,  while  that 
across  the  armature  is  E.  Thus  when  S  =  0,  letting  Et  =  trans- 
former e.m.f.  and  Eu  =  armature  e.m.f., 


Et  =  0,  and  Ea  =  E 


(53) 


FIG.  109. — Two-pole  Model  of  Compensated  Repulsion  Motor. 


It  is  evident  also  that  when  5  =  0  the  flux  through  the 
armature  in  line  with  the  brushes  A  A  will  be  of  zero  value,  so 
that 

<t>t  =  0  (54) 

Let  <j)f  be  the  flux  through  the  armature  in  line  with  the 
brushes  B  B.  This  flux,  neglecting  hysteretic  effects,  is  in  time- 
phase  with  the  line  current  and  produces  by  its  rate  of  change 
through  the  armature  turns  a  counter  e.m.f.  of  value  Ef  =  E, 
giving  to  the  armature  circuit  a  reactance  when  stationary,  of  X. 


216  ALTERNATING  CURRENT  MOTORS. 

The  relation  which  exists  between  the  flux,  the  frequency,  and 
the  number  of  armature  turns  can  be  expressed  thus, 


_  m  ,,,. 

V2~W» 
where  i 

/  =  frequency  in  cycles  per  second 
N  =  effective  number  of  armature  turns 
(j>m  =  maximum  value  of  flux. 

If  C  be  the  actual  number  of  conductors  on  the  armature,  the 

actual  number  of  turns  will  be  —      These  turns  are  evenly  dis- 

2j 

tributed  over  the  surface  of  the  armature,  so  that  any  flux  which 
passes  through  the  armature  core  will  generate  in  each  individual 
turn  an  e.m.f.  proportional  to  the  product  of  the  cosine  of  the 
angle  of  displacement  from  the  position  giving  maximum  e.m.f. 
and  the  value  of  the  maximum  e.m.f.  generated  by  transformer 
action  in  tfye  position  perpendicular  to  the  flux,  or  the  average 

2  C* 

e.m.f.  per  turn  will  be  —  times  the  maximum.     The  —  turns  are 

7T  2 

connected  in  continuous  series,  the  e.m.f.  in  each  half  adding  in 
parallel  to  that  in  the  other  half,  so  that  the  effective  series  turns 


equal—.     Thus,  finally 


(56) 


,  -  The-  .value  of  the  reactance  will  depend  inversely  upon  the  re- 
luctance of  the  paths  through  which  the  armature  current  must 
force  the  flux.  The  major  portion  of  the  reluctance  is  found  in 
the  air-gap,  and  with  continuous  core  material  and  uniform  air- 
gap  around  the  core,  the  reluctance  will  be  practically  constant 
in  ;  all  directions  and  will  be  but  slightly  affected  by  the  change 
in  :  specific  reluctance  of  the  core  material,  provided  magnetic 
saturation  is  not  reached.  In  the  following  discussion  it  will  be 
assumed  that  the  reluctance  is  constant  in  the  direction  of  both 


REPULSION  MOTORS.  217 

sets  of  brushes,  and  that  the  core  material  on  ooth  the  stator  and 
rotor  is  continuous. 

APPARENT  IMPEDANCE  OF  MOTOR  CIRCUITS. 

When  dealing  with  shunt  circuits  it  is  convenient  to  analyze 
the  various  components  of  the  current  at  constant  e.m.f.,  or 
assuming  an  e.m.f.  of  unity,  to  analyze  the  admittance  and  its 
components.  When  series  circuits  are  being  considered,  how- 
ever, the  most  logical  method  is  to  deal  with  the  e.m.fs.  for  con- 
stant current,  or  to  assume  unit  value  of  current  and  analyze 
the  impedance  and  its  various  components.  In  accordance  with 
the  latter  plan,  it  will  be  assumed  initially  that  one  ampere 
flows  through  the  main  motor  circuits  at  all  times  and  the 
various  e.m.fs.  (impedances)  will  thus  be  investigated. 

An  inspection  of  Fig.  109  will  show  that  one  ampere  through 
the  armature  circuit  by  way  of  the  brushes  B  B  will  produce  a 
definite  value  of  'flux  independent  of  any  changes  in  speed  of  the 
rotor,  since  there  is  no  opposing  magnetomotive  force  in  any 
inductively  related  circuit.  From  this  fact  it  follows  that  on 
the  basis  of  unit  line  current  (j>a  has  a  constant  effective  value, 
although  varying  from  instant  to  instant  according  to  an 
assumed  sine  law.  As  will  appear  later,  while  both  the  current 
through  the  armature  and  the  flux  produced  thereby  have  un- 
varying, effective  values  and  phase  positions,  the  apparent  re- 
actance of  the  armature  is  not  constant,  but  follows  a  parabolic 
curve  of  value  with  reference  to  change  in  speed. 

When  the  armature  travels  at  any  certain  speed  the  conductors 
cut  the  flux  which  is  in  line  with  the  brushes  B  B  and  there  is 
generated  at  the  brushes  A  A  an  electromotive  force  proportional 
at  each  instant  to  the  flux  <£/  and  hence  in  time-phase  with  <£f,  or 
with  the  armature  current  through  B  B.  i 

Let  <f>m  =  maximum  value  of  <pf,  then  the  maximum  value  of 
the  e.m.f.  generated  at  A  A  due  to  dynamo  speed  action  will  be, 

V 

(58) 

where  V  is  revolutions  per  second.     The  virtual  value  of  this 
electromotive  force  will  be 


218  ALTERNATING  CURRENT  MOTORS. 

A  comparison  of  (59)  and  (57)  will  show  that  at  a  speed  V 
revolutions  per  second  such  that  V  =  /  in  cycles  per  second, 
Ev  =  Ef  for  any  value  of  (j>m.  Consequently,  the  speed  e.m.f. 
due  to  any  flux  threading  the  armature  turns,  at  synchronism 
becomes  equal  to  the  transformer  e.m.f.  due  to  the  same  flux 
through  the  same  turns.  Ef  is  in  time-quadrature  and  Ev  in 
time-phase  with  the  flux  at  any  speed,  hence,  Ev  is  in  time- 
quadrature  with  Ef  or  in  time-phase  with  the  line  current. 

The  brushes  A  A  remain  at  all  times  connected  directly  to- 
gether by  conductor  of  negligible  resistance  so  that  the  re- 
sultant e.m.f.  between  the  brushes  must  remain  of  zero  value. 
On  this  account  when  an  e.m.f.  Ev  is  generated  between  the 
brushes  by  dynamo  speed  action,  a  current  flows  through  the 
local  circuit  giving  a  magnetomotive  force  such  that  the  flux 
produced  thereby  generates  in  the  armature  conductors  by  its 
rate  of  change,  an  e.m.f.  equal  and  opposite  to  Ev .  This  flux, 
(f>t,  is  proportional  to  Ev  and  being  in  time-quadrature  thereto, 
is  in  time-phase  with  Ef,  or  in  time-quadrature  with  </>/. 

FUNDAMENTAL    EQUATIONS    OF   THE    COMPENSATED   REPULSION 

MOTOR. 

From  the  transformer  relations  it  is  seen  that 

E'  =  ^7F*F     v   .     :;         '       W 

where  \/2  <£/  is  maximum  value  of  flux  due  to  current  through 
brushes  A  A.      See  (57). 

Ev  =  G  </>t  (61) 

where  G  is  a  proportionality  constant. 

Let  S  be  the  speed,  with  synchronism  as  unity,  then 

Ev  =  S  Ef  (62) 

and 

</>/  =  5  <j>f,  (63) 

effective  values  being  used.     This  is  the  fundamental  magnetic 
equation  of  the  compensated  repulsion  motor. 

Flux  </>t  passes  through  the  transformer  turns  on  the  stator  in 
line  with  the  brushes  A  A  as  shown  in  Fig.  109  and  generates 
therein  by  its  rate  of  change  an  e.m.f.  Ef  such  that 

Et  =  n  Ev  (64) 


REPULSION  MOTORS.  219 

where  n  is  the  ratio  of  effective  transformer  to  armature  turns. 
This  e.m.f.  is  in  phase  with  Ev,  in  quadrature  with  Ef  and  hence 
is  in  phase  opposition  with  the  line  current  and  produces  the 
effect  of  apparent  resistance  in  the  main  motor  circuits. 
Combining  (62)  and  (64) 

Et  =  5  n  Ef  (65) 

Since  Ef  is  the  transformer  e.m.f.  in  the  armature  circuits  due 
to  constant  effective  value  of  flux  from  one  ampere,  we  may 
write 

Ef  =  X  (66) 

where  X  is  the  stationary  reactance  of  the  armature  circuit,  so 
that  the  apparent  resistance  of  the  transformer  circuit  is 

R  =  S  n  X  (67) 

Under  speed  conditions  the  armature  conductors  cut  the  flux 
in  line  with  the  brushes  A  A,  and  there  is  generated  thereby  an 
e.m.f.  which  appears  as  a  maximum  at  the  brushes  B  B.  This 
e.m.f.  is  in  phase  with  (/>t,  in  quadrature  with  (f>f  and  in  phase 
opposition  to  Ef.  If  Es  be  the  value  of  this  e.m.f.  we  may 
write 

_  cVtfrS 
x/2108 

from  dynamo  speed  relations.     Comparing   (60)   and   (68)   and 
remembering  that  /  is  unity  in  terms  of  speed,  there  is  obtained 

Es  -  S  Ev  (69) 

from   (62)   and  (66) 

Es  =  S2  Ef  =  S2  X  (70) 

Therefore  the  e.m.f.  across  the  armature  at  B  B  will  be 

Ea  =  Ef-Es  =  X(1-S2)  (71) 

This  e.m.f.  is  in  quadrature  with  the  line  current  and  is  in 
effect  an  apparent  reactance,  so  that  the  apparent  reactance  of 
the  motor  circuits  which  is  confined  to  the  armature  winding  is 

X  =  X(1-S2)  (72) 

The  apparent  impedance  of  the  motor  circuits  at  speed  5  is 


Z_  =  VR*  +  ~X2  =  V(S  n  X)2  +  X2  (1  -  S2)2  (73) 


220  ALTERNATING  CURRENT  MOTORS. 

This  is   the   fundamental   impedance   equation   of   the   ideal 
repulsion-series  motor. 
The  power  factor  is 

R  SnX 

cos  0  =  -=-  =     ,  (74) 

£        V(SnX)2  +  X2  (1-S2)2 

The  line  current  is 

'  =  -  -  E 

Z        VS2n*X2  +  X2  (1-S2)2 

The  power  is, 

P-S'co.g-^****,^.      V     -        (76) 

It  will  be  noted  that  both  the  power  and  the  power  factor  re- 
verse when  5  is  negative.  Thus  the  machine  becomes  a  gen- 
erator when  driven  against  its  torque. 

The  wattless  factor  is, 


Sin  0  =  9-  =     7~— ^~ 

£        \/52  w2  X2  +  X2  (1  -  52)2 

and  becomes  negative  when  5  is  greater  than  1,  so  that  above 
synchronism  when  operated  as  either  a  generator  or  motor  the 
machine  draws  leading  wattless  current  from  the  supply  system. 
At  5  =  1,  Sin  6  =  0,  which  means  that  the  power  factor  is 
unity  at  synchronous  speed,  as  may  be  seen  also  from  eq.  (74). 

At  S  =  0, 


At  S  =  1,       /  =  -7  That  is,  at 

nX 

synchronism  the  line  current  is  equal  to  the  current  at  start 
divided  by  the  ratio  of  transformer  to  armature  turns.  If 
vi  ==  if  the  current  at  synchronism  is  of  the  same  value  as  at 
start  but  the  power  factor  which  at  start  was  0  has  a  value  of  1 
at  synchronism.  This  interesting  feature  will  be  touched  upon 
later. 


REPULSION  MOTORS. 


221 


The  torque  is 


E2nX 


VX2  S2  n2  +  X2  (1  -  S2)2 


=  PnX 


(78) 


and  is  maximum  at  maximum  current  and  retains  its  sign  when 
5  is  reversed. 

When  5  =  0  the  secondary  current,  Is,  is  n  7,  and  is  in  phase 
opposition  with  the  transformer  current  /.  See  Fig.  109.  When 
S  =  1 

and  at  any  speed  S, 


Is  = 


P  +  P 


Speed  in  Percent 
-180      .160      -140       -120      -100        -80        -60       -40         -20          0+20  _  -MO     460      -t-80     +100   +120     -4-140    4160  -f  180 


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Resistance 

.0      -3.2       -2.8      -2.4       -2.0      -l.G       4.2      -.8         -4           0       +.4        j.S        +1.2     4-l.G     4-S.O      +2.4      f2.8     +3.2     +3.< 

FIG.  110. — Characteristics  of  Ideal  Compensated  Repulsion  Motor. 
Is  =  Vn2  P  +  S2  P  = 


X2  (1-S2)2 


(79) 


(80) 


VECTOR  DIAGRAM  OF  COMPENSATED  REPULSION  MOTOR. 

In  Fig.  110  are  shown  the  results  of  calculations  for  a  certain 
ideal  compensated  repulsion  motor  of  which  X  =  1  and.  n-—-%. 
It  is  seen  that  with  speed  as  abscissa,  the  curve  representing 
the  apparent  resistance  of  the.  motor  circuits  is.  a  right-,.. l 


222 


ALTERNATING  CURRENT  MOTORS. 


while  that  for  the  apparent  reactance  is  a  parabola.  At  any 
chosen  speed  the  quadrature  sum  of  these  two  components  gives 
the  apparent  impedance  of  the  motor.  Since  the  scale  for  rep- 
resenting the  speed  is  in  all  respects  independent  of  that  used 
for  the  apparent  resistance,  it  is  possible  always  so  to  select  values 
for  the  one  scale  such  that  a  given  distance  from  the  origin  may 
simultaneously  represent  both  the  resistance  and  the  speed. 
This  method  of  plotting  the  values  leads  to  a  very  simple  vector 
diagram  for  representing  both  the  value  and  phase  position  of  the 
apparent  impedance  at  any  speed,  and  for  determining  the 


-180     -160      -140 


Speed  in  Percent 
-20          0 


FIG.  111. — Characteristics  of  Ideal  Compensated  Repulsion  Motor. 

power-factor  from  inspection.  Thus  at  any  speed  such  as  is 
shown  at  G  the  distance  O  G  is  the  apparent  resistance,  the  dis- 
tance G  P  is  the  apparent  reactance,  O  P  is  the  apparent  im- 
pedance while  the  angle  P  O  G  is  the  angle  of  lead  of  the  pri- 
mary current  and  its  cosine  is  the  power-factor. 

Fig.  Ill  gives  the  complete  performance  characteristics  of  the 
above  ideal  compensated  repulsion  motor  at  various  positive 
and  negative  speeds  when  operated  at  an  impressed  e.m.f.  of 
100  volts. 

.  It  will  be  noted  that  the  armature  e.m.f.,  which  has  a  certain 
value  at  standstill,  decreases  with  increase  of  speed,  becomes 


REPULSION  MOTORS.  223 

zero  at  synchronism  and  then  increases  at  higher  speeds.  The 
transformer  e.m.f.  is  zero  at  starting,  increases  to  a  maximum 
at  synchronism  and  then  continually  decreases  with  increase  of 
speed. 

CALCULATED  PERFORMANCE  OF  COMPENSATED  REPULSION 

MOTOR. 

The  inductive  portion  of  the  impedance  is  contained  wholly 
by  the  armature  circuit,  while  the  non-inductive  is  confined  to 
the  transformer  coil;  thus  the  power-factor  is  zero  at  standstill, 
reaches  unity  at  synchronism  and  then  decreases  due  to  the 
lagging  component  of  the  motor  impedance  (leading  wattless 
current).  In  comparison  with  the  ordinary  compensated  series 
motor  whose  armature  e.m.f.  is,  for  the  most  part,  non-inductive 
and  continually  increases  with  increase  of  speed,  and  whose  in- 
ductive field  e.m.f.  decreases  continually  with  increase  of  speed 
and  whose  power-factor  never  reaches  unity,  the  compensated- 
repulsion  motor  furnishes  a  most  striking  contrast.  The  machine 
resembles  the  repulsion  motor  in  regard  to  its  magnetic  behavior, 
but  the  performance  of  its  electric  circuits  differs  from  that  of  the 
repulsion  motor  due  to  the  fact  that  the  speed  e.m.f.  introduced 
into  the  armature  circuit  B  B  (Fig.  109)  which  has  been  sub- 
stituted for  the  field  coil  of  the  repulsion  motor  (see  Fig.  106) 
is  in  a  direction  continually  to  decrease  the  apparent  reactance 
of  the  field  circuit  and  thus  to  decrease  the  inductive  component 
of  the  impedance  of  the  circuits  and  to  improve  the  power  factor 
and  the  operating  characteristics.  It  is  an  interesting  fact  that 
under  all  conditions  of  operation  the  e.m.f.  in  the  coils  short 
circuited  by  the  brushes  B  B  is  of  zero  value,  so  that  no  ob- 
jectional  features  are  introduced  by  substituting  the  armature 
circuit  for  the  field  coil  of  the  repulsion  motor,  while  the  per- 
formance is  materially  improved.  Experiments  show  that  even 
with  currents  of  many  times  normal  value  and  at  the  highest 
commercial  frequency  no  indication  of  sparking  is  found  at  the 
brushes  B  B.  This  feature  will  be  treated  in  detail  later. 

An  inspection  of  Fig.  110  and  of  equation  (73)  will  reveal  the 
fact  that  at  synchronism  the  apparent  impedance  is  n  times  its 
value  at  standstill.  If  n  be  made  unity,  the  apparent  impedance 
at  synchronism  will  be  equal  to  that  at  standstill,  while  between 
these  speeds  it  varies  inappreciably.  This  means  that  from  zero 


224  ALTERNATING  CURRENT  MOTORS. 

speed  to  synchronism  the  primary  current  varies  but  slightly, 
and  that  the  torque,  which  is  proportional  to  the  square  of  the 
primary  current  is  practically  constant  throughout  this  range  of 
speed.  These  facts  show  that  a  unity  ratio  compensated-repulsion 
motor  is  a  constant  torque  machine  at  speeds  from  negative  to 
positive  synchronism,  the  relative  phase  position  of  the  current 
and  the  e.m.f.  changing  so  as  always  to  cause  them  to  give  by 
their  vector  product  the  power  represented  by  the  torque  at  the 
various  speeds.  Above  synchronism  the  torque  decreases  con- 
tinually, tending  to  disappear  at  infinite  speed. 

Any  desired  torque-speed  characteristic  within  limits  can  be 
obtained  by  giving  to  n  a  corresponding  value,  the  torque  at 
synchronism  being  equal  to  the  starting  torque  divided  by  the 
square  of  the  ratio  of  transformer  to  armature  turns. 

In  connection  with  the  discussion  of  the  expression  for  deter- 
mining the  value  of  the  torque  it  is  well  to  mention  the  fact  that 
the  commonly  accepted  explanations  as  to  the  physical  phe- 
nomena involved  in  the  production  of  torque  must  be  somewhat 
modified  if  actual  conditions  of  operation  known  to  exist  are  to 
be  represented.  Referring  to  Fig.  109,  it  will  be  noted  that 
when  the  armature  is  stationary  there  exists  no  magnetism  in 
line  with  the  brushes  A  A,  so  that  the  current  which  enters  the 
armature  by  way  of  the  brushes  B  B  could  not  be  said  to  pro- 
duce torque  by  its  product  with  magnetism  in  mechanical 
quadrature  with  it.  Similarly,  the  flux  in  line  with  the  brushes 
B  B  could  not  be  said  to  be  attracted  or  repelled  by  magnetism 
which  does  not  exist.  That  the  current  through  A  A  produces 
torque  by  its  product  with  the  magnetism  due  to  current  through 
B  B  would  be  contrary  to  accepted  methods  of  reasoning,  since 
both  currents  flow  in  the  same  structure,  yet,  as  concerns  the 
torque,  the  effect  is  quite  the  same  as  though  the  flux  in  line 
with  the  brushes  B  B  were  due  to  current  in  a  coil  located  on 
the  field  core.  (As  shown  in  Fig.  106  for  the  ordinary  repulsion 
motor.)  These  fundamental  facts  were  discussed  in  the  chapter 
on  electromagnetic  torque.  See  Fig.  98. 

OBSERVED  PERFORMANCE  OF  COMPENSATED  REPULSION  MOTOR. 

The  calculated  impedance  characteristics  shown  in  Fig.  1 10  are 
based  on  arbitrarily  assumed  constants  of  a  repulsion-series 
motor  under  ideal  conditions.  It  is  obviously  impossible  to  ob- 


REPULSION  MOTORS. 


225 


tain  such  characteristics  from  an  actual  motor,  since  all  losses 
and  minor  disturbing  influences  have  been  neglected  in  deter- 
mining the  various  values.  As  a  check  upon  the  theory  given 
above,  the  curves  of  Figs.  112  and  113  as  obtained  from  tests  of  a 
compensated-repulsion  motor,  are  presented  herewith.  It  will 
be  observed  that  the  apparent  resistance  of  the  transformer  coil 
varies  directly  with  the  speed  and  becomes  negative  at  negative 
speed,  while  the  apparent  reactance  of  the  armature  decreases 
with  increase  of  speed  in  either  direction  and,  following  approx- 


-18   -16  -14    -12    -10    _! 


Speed  in  100  R.P.  M. 

6    -4      -2        0     4-2     -1-4 


+8     -1-10   -t-12    <-14   ^-16   +18 


X 


\ 


'I 

-1 

-3 
-3 


-5 


FIG.  112. — Test  of  Compensated  Repulsion  Motor — Active 
Factors  of  Operation. 

imately  a  parabolic  law,  reverses  and  becomes  negative  at  speeds 
slightly  in  excess  of  synchronism.  A  comparison  of  the  general 
shape  of  the  curves  of  Fig.  112  and  Fig.  110  will  show  to  what 
extent  the  assumed  ideal  conditions  can  be  realized  in  practice, 
and  it  would  indicate  that,  as  concerns  the  active  factors  of  opera- 
tion, the  equations  given  represent  the  facts  involved.  The 
neglect  of  the  local  resistance  of  the  transformer  circuit  leads  to 
the  discrepancy  between  the  theoretical  and  observed  curves  as 
found  at  zero  speed,  the  latter  curve  indicating  a  certain  apparent 
resistance  when  the  armature  is  stationary.  Similarly  at  syn- 


226 


ALTERNATING  CURRENT  MOTORS. 


chronous  speed  the  observed  apparent  reactance  of  the  arma- 
ture is  not  of  zero  value  due  to  the  local  leakage  reactance  of 
the  circuit. 

In  the  determination  of  the  theoretical  curves  only  active 
factors  have  been  considered,  and  it  has  been  shown  that  the 
apparent  reactance  of  the  motor  circuits  is  confined  to  the  arma- 
ture, while  the  e.m.f.  counter  generated  in  the  transformer  coil 
gives  the  effect  of  apparent  resistance  located  exclusively  within 
this  coil.  The  neglected  disturbing  factors,  the  apparent  re- 
sistance of  the  armature  and  the  apparent  reactance  of  the 
transformer,  are  of  relatively  small  and  practically  constant 


-18      .16       -14 


FIG.    113. — Test   of  Compensated    Repulsion   Motor — Dis- 
turbing Factors  of  Operation. 

value  throughout  the  operating  range  of  speed  from  negative  to 
positive  synchronism,  but  they  become  of  prime  importance 
when  the  speed  exceeds  this  value  in  either  direction,  as  shown 
by  the  curves  of  Fig.  113  obtained  from  the  test  of  a  com- 
pensated repulsion  motor  giving  the  curves  of  Fig.  112.  Th° 
predominating  influence  of  the  disturbing  factors  above  syn 
chronism  is  attributable  largely  to  the  effect  of  the  short  circuit 
by  the  brushes  .4  A  (Fig.  109)  of  coils  in  which  there  is  pro- 
duced an  active  e.m.f.  by  combined  transformer  and  speed 
action.  This  short  circuiting  effect  will  be  treated  in  detail 
later. 


REPULSION  MOTORS.  227 

CORRECTIONS  FOR  RESISTANCE  AND  LOCAL  LEAKAGE  REACTANCE. 

The  resistance  and  local  leakage  reactance  of  the  coils  may 
be  included  in  the  theoretical  equations  as  follows: 
Let  Rt    =  resistance  of  transformer  coil 
Ra  =  resistance  of  armature  circuit 
Rs   =  resistance  of  secondary  circuit 
Xt  =  leakage  reactance  of  transformer 
Xa  =  leakage  reactance  of  armature 
Xs  =  leakage  reactance  of  secondary  circuit 
then  copper  loss  of  motor  circuits  will  be 

(81) 
(82) 


P  [Rt  +  Ra  +  (n2  +  S2)  Rs]  =  P  Rm  (83) 

where  Rm  is  the  effective  equivalent  value  of  the  motor  circuit 
resistance,  that  is, 

R  m  =  Rt  +  Ra  +  (n2  +  S2)  Rs  (84) 

Similarly  it  may  be  shown  that  the  effective  equivalent  value 
of  the  leakage  reactance  of  the  motor  circuits  is 

Xm  -  Xt  +  Xa  +  (n2  +  S2)  Xs  (85) 

combining  equations  (84)  and  (85)  with  (73)  the  expression  for 
the  apparent  impedance  of  the  motor  circuits  becomes 


(86) 


V[S  n 


[X  (1  -  S2)  +  Xt  +  Xa  +  (n2  +  S2)  X  J2 

E 

"  \/(S  n  X  +  Rm)*  +  [X  (i  _S2)  +  xw]2  (87) 


m 

cos  0  =  m  C88) 

\/[X(l-S2)-fXm]2+  (S 


Input  =  E  I  cos  0  (89) 

Output  =  E  I  cos  6  -  P  Rm  =  P  (90) 

E*(SnX+Rm) 


228  ALTERNATING  CURRENT  MOTORS. 

1  _  C2\     I     V     12  (91) 


=  -  -22       PSnX 


torque  =  D  =  -        =  P  n  X  (93) 

The  above  equations,  though  incomplete  on  account  of  ne- 
glecting the  brush  shortening  effect  and  the  magnetic  losses  in 
the  cores,  represent  quite  closely  the  electrical  characteristics 
of  the  compensated  repulsion  motor  when  operated  between 
negative  and  positive  synchronism,  throughout  which  range  of 
speed  the  disturbing  factors  are  of  secondary  importance. 

BRUSH  SHORT-CIRCUITING  EFFECT. 

The  e.m.f.  in  the  coils  short  circuited  by  the  brushes  can  be 
treated  by  a  method  similar  to  that  used  with  the  repulsion 
motor.  Referring  to  Fig.  109,  the  coil  under  the  brush  A  is 
subjected  to  the  transformer  effect  of  the  flux,  <£f,  in  line  with 
the  brushes,  B  B,  and  the  dynamo  speed  effect  of  the  flux,  <j)t, 
in  line  with  the  brushes  A  A. 

Effective  values  being  used  throughout,  the  transformer  e.m.f. 
will  be,  assuming  C  actual  conductors  on  the  armature, 


€t 


x/2"        108 


in  volts  for  one  coil.     See  equation  (57).     This  e.m.f.  is  in  time- 
quadrature  with  </>/. 

The  dynamo  speed  e.m.f.  in  volts  for  one  coil  will  be, 


See  equation  (59).  This  e.m.f.  is  in  time-phase  with  <£/  and 
hence  is  time-quadrature  with  <£/.  Thus  the  electromotive  force 
in  the  coil  under  the  brush  A  is 


REPULSION  MOTORS.  229 

v^,)  (96) 


But  V  =  /  5  and  0,  =  S  <j>f  (97) 

See  equation  (63),  hence 

(98) 


so  that 

£«  =  ^f  (l-S*)         •••-     (99) 

This  resultant  electromotive  force  has  a  value  at  standstill 
when  5  is  zero,  of 

7T    ,-, 

2Ef      InX 

-£-     Tc  <10°) 

4 
See  equation  (66).     Thus 

finally,  Ea  =  I-^-  (1  -  S2)  (101) 

When  the  armature  is  stationary  the  electromotive  force  in 
the  coil  short  circuited  by  the  brush  A  has  the  value  given  by 
equation  (100),  which,  with  any  practical  motor,  is  of  sufficient 
value  to  cause  considerable  heating  if  the  armature  remains  at 
rest,  or  to  produce  a  fair  amount  of  sparking  as  the  armature 
starts  in  motion.  At  synchronous  speed,  however,  this  electro- 
motive force  disappears  entirely,  and  the  performance  of  the 
machine  as  to  commutation  is  perfect.  As  the  speed  exceeds 
this  critical  value  in  either  the  positive  or  negative  direction, 
the  electromotive  force  in  the  short-circuited  coil  increases  rap- 
idly, resulting  in  a  return  in  an  augmented  form  of  the  sparking 
found  at  lower  -  speeds  and  producing  the  disturbing  factors 
shown  by  the  curves  of  Fig.  113. 

Since  the  e.m.f.  in  the  coil'  under  the  brush  A  reduces  to  zero 
at  both  positive  and  negative  synchronism  and  reverses  with 
reference  to  the  time-phase  position  of  the  line  current  at  speeds 
exceeding  synchronism  in  either  direction,  it  possesses  at  high 
speeds  the  same  time-phase  position  when  the  machine  is  oper- 
ated as  a  generator  as  when  it  is  used  as  a  motor.  The  time- 
phase  of  its  reactive  effect  upon  the  current  which  flows  in  the 


230  ALTERNATING  CURRENT  MOTORS. 

armature  through  the  brushes  B  B  is  of  the  same  sign  at  high 
positive  and  negative  speeds,  but  reversed  from  the  phase  posi- 
tion of  the  effect  at  speeds  below  synchronism.  A  study  of  the 
test  curves  of  Fig.  113  will  show  the  magnitude  of  these  effects, 
and  the  reversal  of  their  time-phase  positions  in  accordance 
with  the  theoretical  considerations. 

With  reversal  of  direction  of  rotation  the  time-phase  position 
of  the  flux  threading  the  transformer  coil  (Fig.  109)  reverses 
with  reference  to  the  line  current,  and  hence  in  its  reactive 
effect  upon  the  transformer  flux  the  current  in  the  coil  short 
circuited  by  the  brush  A  becomes  negative  at  speeds  above 
negative  synchronism,  though  positive  above  synchronism  in 
the  positive  direction.  At  speeds  below  synchronism,  when  the 
flux  is  large  the  e.m.f.  is  small,  and  vice  versa,  so  that  the  reactive 
effect  is  in  any  case  relatively  small  and  of  more  or  less  con- 
stant value.  See  Fig.  113. 

It  will  be  noted  that  in  analyzing  the  disturbing  factors  no  ac- 
count has  been  taken  of  the  short-circuiting  effect  at  the  brushes 
B  B,  Fig.  109.  This  treatment  is  in  accord  with  the  statement 
previously  made  that  the  component  e.m.fs.  generated  in  the 
coils  under  these  brushes  are  at  all  times  of  values  such  as  to 
render  the  resultant  zero.  The  proof  of  this  fact  is  as  follows: 

The  transformer  e.m.f.  in  the  coil  under  B  due  to  flux,  <f>t,  in 
lines  with  brushes  A  A  is 

"-OT  (102) 

See  equation  (94).     This  e.m.f.  is  in  time-quadrature  with  <f>t. 
The  dynamo  speed  e.m.f.  is 

7T   V  6f 


This  e.m.f.  is  in  time-phase  with  <f>f,  in  time  quadrature  with 
>  and  is  in  phase  opposition  to  ef.     Thus  the  resultant  e.m.f.  is 


Eb  =  ef  -  ev  =  g-o*  (/  fa  -  V  <£f)  (104) 

Since  V  =  /Sand<£,  =  S  <j>f  from  equations  (97)  and  (63), 

fft-Vfo  (105) 

and 

Eb  =  0  (106) 


REPULSION  MOTORS.  231 

This  theoretical  deduction  is  substantially  corroborated  by 
experimental  evidence,  as  has  been  noted  above.  Even  upon 
superficial  examination  such  a  result  is  to  be  expected,  since  the 
vector  sum  of  all  e.m.fs.  in  the  armature  in  mechanical  line  with 
the  short-circuited  brushes  A  A  must  be  zero,  while  the  e.m.f. 
in  the  coil  at  brush  B  must  equal  its  proper  share  of  this  e.m.f. 
or 

Eb  =  £f~    =  °  (107) 

A  similar  course  of  reasoning  allows  of  the  determination  of 
the  electromotive  force  under  the  brush  A.  See  equation  (71). 

-* 


a  ~~    C2    '  2'C 

for  unit  current.     For  7  amperes  this  becomes 

Ea  =  I~¥cT(l~s*)  (109) 

See  equation  (101). 

From  the  facts  just  indicated  it  would  seem  that  perfect  com- 
mutation dictates  that  the  electromotive  force  across  a  diameter 
ninety  electrical  degrees  from  the  brushes  upon  the  armature  be 
at  all  times  of  zero  value. 

It  has  been  stated  that  the  magnetic  circuits  of  the  compen- 
sated repulsion-series  motor  are  quite  the  same  as  those  of  the 
repulsion  motor.  The  fluxes  in  line  with  the  two  brush  circuits 
under  all  conditions  are  in  time-quadrature  and  have  relative 
values  varying  with  the  speed  such  that  at  all  times 

*>->$*  (HO) 

There  exists,  therefore,  at  all  speeds  a  revolving  magnetic  field 
elliptical  in  form  as  to  space  representation.  At  standstill  the 
ellipse  becomes  a  straight  line  in  the  direction  of  the  brushes  B  B 
(Fig.  109),  at  infinite  speed  in  either  direction  the  ellipse  would 
again  be  a  straight  line  in  the  direction  of  the  brushes  A  A  ,  while 
at  either  positive  or  negative  synchronism  the  ellipse  is  a  true 
circle,  the  instantaneous  maximum  value  of  the  revolving  mag- 
netism traveling  in  the  direction  of  motion  of  the  armature.  At 
synchronous  speed,  therefore,  the  magnetic  losses  in  the  arma- 
ture core  disappear,  while  the  losses  in  the  stator  core  are  evenly 
distributed  around  its  circumference. 


CHAPTER  XV. 

MOTORS  OF  THE  SERIES  TYPE  TREATED  BOTH  GRAPHICALLY 
AND  ALGEBRAICALLY. 

.    THE  PLAIN  SERIES  MOTOR. 

The  combined  transformer  and  motor  features  of  commutator 
type  of  alternating  current  machinery  are  well  exemplified  in  the 
plain  series  motor  as  illustrated  in  Fig.  114.  When  the  rotor  is 
stationary,  the  field  and  armature  circuits  of  the  motor  form  two 
impedances  in  series.  Assuming  initially  an  ideal  motor  without 
resistance  and  local  leakage  reactance,  each  impedance  consists 
of  pure  reactance,  the  current  in  the  circuit  having  a  value  such 
that  its  magnetomotive  force  when  flowing  through  the  arma- 
ture and  field  turns  causes  to  flow  through  the  reluctance  of  the 
magnetic  path  that  value  of  flux  the  rate  of  change  of  which 
generates  in  the  windings  an  electromotive  force  equal  to  the 
impressed. 

If  E  be  the  impressed  e.m.f.,  Ef  the  counter  transformer  e.m.f. 
across  the  field  coil  and  Ea  the  counter  transformer  e.m.f.  across 
the  armature  coil,  when  the  armature  is  stationary 

E  =  £/  +  £„  (111) 

From  fundamental  transformer  relations  there  is  obtained  the 
equation 

'  see  eq.  (55)  (112) 


where    /  =  frequency  in  cycles  per  second 
Nf  =  effective  number  of  field  turns 
<j)f  =  maximum  value  of  field  flux. 
Similarly 


_  q 

\/2l08 


where  Na  *-:  effective  number  of  armature  turns 
$a  =  maximum  value  of  armature  flux. 
232 


SERIES  MOTORS. 


233 


FUNDAMENTAL  EQUATIONS   OF  SERIES   MOTOR  WITH   UNIFORM 

AIR-GAP  RELUCTANCE. 
Since  the  field  and  armature  circuits  are  electrically  series  con- 


Speed  E.M.F. 


FIG.  114. — Circuit  and  Vector  Diagrams  of  Plain  Series  Motor. 

nected  and  are  mechanically  so  placed  as  not  to  be  inductively 
related,  with  uniform  reluctance  around  the  air  gap  the  fluxes  in 
mechanical  line  with  the  two  circuits  being  due  to  the  magneto- 


234  ALTERNATING  CURRENT  MOTORS. 

motive  force  of  the  same  current  will  be  proportional  to  the 
effective  number  of  turns  on  the  two  circuits. 
Therefore 


If  n  be  the  ratio  of  effective  field  to  armature  turns 

Nf~HNm  (115) 


and 


Let  C  be  the  actual  number  of  conductors  on  the  armature, 
then 

Na  -  ^  (see  eq.  56)  (117) 

Under  speed  conditions  the  armature  conductors  cut  the  field 
magnetism  and  there  is  generated  by  dynamo  action  a  counter 
e.m.f.  proportional  to  the  product  of  the  field  flux  and  the  speed 
in  time-phase  with  the  flux,  in  leading  time  quadrature  with  the 
field  e.m.f.,  Ef  and  the  armature  e.m.f.  Ea  and  in  phase  opposi- 
tion with  the  current. 

Thus 

Ev  =  v$fo8  (see  eq'  59) 

where  V  is  revolutions  per  second  of  bipolar  model. 
Combining  (117)  and  (118) 


If  S  be  the  speed  with  synchronism  as  unity,  then 

V  =  Sf  (120) 

and 

.   E^27^w^  (121) 

combining  (113),   (116)  and  (121) 

TT     <?JW 

Sn  (122) 


SERIES  MOTORS.  235 

combining  (112),   (115)  and   (121) 

Ef  S  Na  _  Ef  S 

Ev  '    HvT"       ~ir 

comparing  (122)  and  (123) 

Ef  =  n~  Ea 

Under  speed  conditions  the  impressed  e.m.f.  is  balanced  by 
three  components,  Ev  in  time  phase  opposition  with  the  line  cur- 
rent and  Ef  and  Ea,  both  in  leading  time  quadrature  with  the 
line  current. 

Thus  _ 

(125) 


S*nz+(Ea  +  n*Ea)*  =  £«  VS2  n?+(l  +  n2)*  (126) 

This  is  the  fundamental  electromotive  force  equation  of  the  plain 
series  motor  having  uniform  reluctance  around  the  air-gap. 

On  the  basis  of  unit  line  current  the  electromotive  forces  may 
be  treated  as  impedances,  as  was  done  with  the  repulsion-series 
motor,  so  that  the  impedance  equation  becomes 

Z  -  Xa  \/S2n2+  (1+w2)2  (127) 

where  S  n  Xa  =  R_  and  (1  -f  n2}  Xa  =  _X 

The  power  factor  is 

- 


(128) 
w2)2 

which  reverses  when  5  becomes  negative  and  continually  ap- 
proaches unity  with  increase  of  S  in  either  direction. 

When  5  =  1,  or  at  synchronism  I 

cos  0  =      .    ---------------  ......  _  (129) 


which  when  n  =  1  or  for  unity  ratio  of  field  to  armature  turns 
becomes 


go--447-  (130) 

and  decreases  with  either  an  increase  or  decrease  of  n.     It  is 


236  ALTERNATING  CURRENT  MOTORS. 

apparent  "therefore  that  the  power  factor  of  such  a  machine  is 
inherently  very  low  and  cannot  be  improved  by  a  mere  change 
in  the  ratio  of  field  to  armature  turns. 
The  line  current  is 


Z"    =  X' 


The  power  is 

P  =  £/cosfl  =  £-^2+5J+My    -          (132) 

which  becomes  negative  when  5  reverses,  or  the  machine  oper- 
ates as  a  generator  when  driven  against  its  natural  tendency  to 
rotation. 

The  torque  is 


which  is  maximum  at  maximum  current  and  retains  its  sign 
when  5  is  reversed. 

At  starting  the  torque  is 

Do  _£.__£_  •  '     (134) 

At  synchronous  speed,  the  torque  is 

g'"f.;n' +("+»')'  (135) 

and 

Ds          (l  +  «2)2 


(136) 

which  when  n  is  negligibly  small  approaches  a  value  of  unity  and 
when  n  is  infinitely  large  also  tends  to  reach  a  value  of  unity. 
When  n  =  I  equation  (136)  reduces  to 

g|_  rrw  =  -8  ,  '    <137> 

the  interpretation  of  which  is  that  the  torque  of  the  unity-ratio 
single-phase,  plain  series  motor  with  uniform  reluctance  around 


SERIES  MOTORS.  237 

the  air-gap  varies  only  20  per  cent,  from  standstill  to  syncnfo'- 
nism,  and  therefore,  that  such  a  machine  is  imsuited  for  traction. 
This  statement  applies  to  the  ideal  single-phase  motor  without 
internal  losses,  and  must  be  somewhat  modified  to  include  true 
operating  conditions.  The  method  of  treating  the  various  losses 
has  previously  been  discussed  and  will  further  be  enlarged  upon 
in  connection  with  the  compensated  types  of  series  machines.  A 
little  consideration  will  show  that  such  modifications  as  must  be 
introduced  have  a  detrimental  effect  upon  the  characteristics  of 
the  machine,  and  tend  to  lay  greater  stress  upon  the  statement 
.just  made.  These  facts  are  graphically  represented  in  the  per- 
formance (impedance)  diagram  of  Fig.  114.  O  A  is  the  power 
and  A  B  the  reactive  components  of  the  apparent  field  impedance 
at  starting  while  B  C  and  C  D  are  the  corresponding  power  and 
reactive  components  of  the  apparent  armature  impedance.  The 
power  component  of  apparent  armature  impedance  due  to 
dynamo  speed  action  is  shown  as  D  E  or  D  F,  giving  the  resultant 
impedance  under  speed  conditions  of  O  E  or  0  F  and  indicating 
an  angle  of  lag  of  the  circuit  current  behind  the  impressed  e.m.f. 
of  E  0  A  or  F  O  A  .  The  variation  in  torque  due  to  increase  of 
speed  from  synchronism  to  double  synchronism  with  a  unity 
ratio  constant  reluctance  machine,  as  represented  in  Fig.  114, 
would  be  as  the  square  of  the  ratio  of  0  F  to  O  E. 

FUNDAMENTAL  EQUATIONS  FOR  MOTOR  WITH  NON-UNIFORM  RE- 

LUCTANCE. 

An  inspection  of  equation  (136)  will  reveal  the  fact  that  a 
change  in  the  value  of  n  does  not  improve  the  torque  charac- 
teristics of  the  machine  unless  such  change  be  accompanied  with 
an  increase  in  reluctance  of  the  magnetic  structure  in  line  with 
the  brushes  Bl  B2  (Fig.  114).  That  is  to  say,  if  the  mechanical 
construction  is  such  that  equation  (114)  may  be  written 


where  m  is  a  constant  of  a  value  many  times  unity,  the  oper- 
ating characteristics  of  the  machine  become  much  improved. 
Thus  equation  (116)  becomes  ,,,, 


=  m  n 


238  ALTERNATING  CURRENT  MOTORS. 

and  equation  (122)  is  changed  to 


E  =  y/E.'+VS.  +  E.y-  +  +  E,  (142) 


Cos0=    ~ 


<" 


when  5  =  1  or  at  synchronism 

1_ 

n 1 

Cos  <?  =      |   .  fv  ;       7"~r2Tr JV2  =      I          /mn2-4-1\2  (146) 


I 

\ 


With  an  excessively  large  reluctance  of  the  magnetic  structure 
in  line  with  the  brushes  B1B2  (Fig.  114),  that  is,  with  an  enor- 
mous value  of  m,  the  power  factor  at  synchronous  speed  ap- 
proaches 

Cos  0  =  -7*-  -  (147) 


the  interpretation  of  which  equation  is  that  the  operating  power 
factor  of  such  a  machine  is  largely  dependent  upon  the  ratio  of 
field  to  armature  turns.  A  little  study  will  show  that  at  any 
chosen  speed,  whether  synchronous  or  not,  the  cotangent  of  the 
angle  of  lag  is  directly  proportional  to  the  ratio  of  armature  to 
field  turns,  and  that  the  power-factor,  the  corresponding  cosine, 


SERIES  MOTORS.  239 

can  be  given  any  desired  value  by  a  proper  proportioning  of  the 
windings.  This  feature  will  be  treated  more  in  detail  when  deal- 
ing with  compensated  motors. 

The  current  of  the  high  brush-line-reluctance  machine  is 

E   __  E  n  I 

7  '    "  (148) 


The  power  is 

E2  n  S 


P  =  E  I  cos  0  = 


. 

Xf        „,       /m  n2H  1 


/mn 
\    mn 

The  torque  is 


S  Xf    '  /m 


At  starting  the  torque  is 


At  synchronous  speed  the  torque  is 


(152) 


which  ratio,  with  an  enormous  value  of  m,  approaches 
D, 


the  significance  of  which  is  that  the  change  of  torque  from  stand- 
still to  synchronism  can  be  altered  at  will  by  change  in  the  ratio 
of  field  to  armature  turn  and  that  a  relatively  low  value  of  n 
would  produce  a  machine  suitable  for  traction. 


240 


ALTERNATING  CURRENT  MOTORS. 


By  using  projecting  field  poles  thus  leaving  large  air-gaps  in 
the  axial  brush  line  and  thereby  increasing  the  reluctance  of  the 
structure  in  line  with  the  magnetomotive  force  of  the  armature 
current,  the  flux  produced  by  the  armature  current  may  be  ma- 
terially reduced,  thus  giving  to  m  a  relatively  large  value,  and 

Field  Core 


Field  Core 


Speed  E.M.F. 


FIG.   115. — Circuit  and  Vector  Diagrams  of  Inductively- 
Compensated  Series  Motor. 

the  power  factor  will  be  thereby  correspondingly  increased  with 
a  resultant  improvement  in  the  torque  characteristics  of  the 
machine.  Even  under  the  most  favorable  conditions,  however, 
it  is  impossible  to  reduce  the  reactance  of  the  armature  circuit 
to  an  inappreciable  value,  that  is,  to  give  to  m  an  enormous 


SERIES  MOTORS. 


241 


value,  due  to  the  inevitable  presence  of  the  magnetic  material 
of  the  projecting  poles. 

INDUCTIVELY  COMPENSATED  SERIES  MOTOR. 

The  most  satisfactory  method  of  reducing  the  inductive  effect 
of  the  armature  current  is  to  surround  the  revolving  armature 


Field  Coil 


Field  Coil 


Speed  E.M.F. 


of  Current 


O  A 

FIG.  116. — Circuit  and  Vector  Diagrams  of  Conductively 
Compensated  Series  Motor. 

winding  with  properly  disposed  stationary  conductors  through 
which  current  flows  equal  in  magnetomotive  force  and  opposite 
in  phase  to  the  current  in  the  armature.  This  compensating 
current  may  be  produced  inductively  by  using  the  stationary 


242  ALTERNATING  CURRENT  MOTORS. 

winding  as  the  short  circuited  secondary  of  a  transformer  of 
which  the  armature  is  the  primary,  as  illustrated  diagrammat- 
ically  in  Fig.  115,  or  the  main  line  current  may  be  sent  di- 
rectly through  the  compensating  coil  as  shown  in  Fig.  116.  In 
the  former  case  the  transformer  action  is  such  that  the  com- 
pensation is  practically  complete,  giving  minimum  combined 
reactance  of  the  two  circuits  while  in  the  latter  case,  the  propor- 
tion of  compensation  can  be  varied  at  will.  It  is  found  that  in 
any  case  the  best  general  effects  are  produced  when  the  com- 
pensation is  complete,  and  experiments  seem  to  indicate  that 
under  such  conditions  the  two  methods  of  compensation  differ 
inappreciably  for  strictly  alternating  current  work,  but  that 
for  direct  current  operation  where  the  forced  compensation  can 
be  used  to  prevent  field  distortion  and  to  improve  the  commuta- 
tion, the  latter  method  is  preferable. 

CONDUCTIVELY   COMPENSATED   SERIES   MOTOR. 

Referring  to  Figs.  115  and  116,  assume  an  ideal  series  motor 
with  complete  compensation,  letting  n  be  the  ratio  of  effective 
field  to  armature  turns;  at  any  speed  S  with  synchronism  as 
unity,  the  apparent  impedance  of  the  motor  circuits  will  be 

IS* (154) 

z-x^g+i 

of  which 

X  =  Xf  (155) 

represents  the  reactance  of  the  motor  circuits  which  is  confined 
to  the  field  coil,  and  of  which 

(156) 

represents  the  apparent  resistance  effect  of  the  dynamo  speed 
e.m.f.  counter  generated  at  the  brushes  Bl  B2  due  to  the  cutting 
of  the  field  flux  by  the  armature  conductors  (see  eq.  123). 
The  power  factor  is 

S 

(157) 


which  continually  approaches  positive  or  negative  unity  with 
increase  of  speed  in  the  corresponding  direction. 


SERIES  MOTORS.  243 

At  synchronism  when  5=1,  the  power  factor  is 

cos  6  =  ~~  (see  eq.  147)  (158) 

V  H-w2 

The  line  current  is 

•      '-K'TfT,       •       <15* 

The  power  is  5 

P  =  £/cos0  =  J.^ 
The  torque  is  «2 

P  E^  ] 


The  ratio  of  the  torque  at  synchronous  speed  to  that  at  stand- 
still is 

(162) 


which  in  a  practical  machine  can  be  made  as  much  smaller  than 
unity  as  desired  by  a  proper  proportioning  of  the  field  and  arma- 
ture windings.  It  is  evident,  therefore,  that  such  a  machine  can 
be  made  suitable  for  traction  when  a  proper  value  of  n  is  chosen. 

COMPLETE  PERFORMANCE  EQUATIONS  OF  COMPENSATED  MOTORS. 

The  above  equations  refer  to  ideal  motors  without  resistance 
and  local  leakage  reactance  and  devoid  of  all  minor  disturbing 
influences.  A  close  approximation  for  the  effect  of  the  resist- 
ance and  leakage  reactance  may  be  obtained  as  follows: 

Let  rf  =  resistance  of  field  coil 

rc  =  resistance  of  compensating  coil  (reduced  to  a  1  to  1, 

armature  ratio) 
ra  =  resistance  of  armature 
Xf  =  local  reactance  of  field  coil 
xa  =  combined  leakage  reactance  effect  of  armature  and 

compensating  coils 


244  ALTERNATING  CURRENT  MOTORS. 

Then  the  apparent  impedance  is 

z  =  \  HT*  +  r/  +  Tc  +  r°)  +  (Xf  +  Xi  +*°)2 

Power  factor  is 

r,  —  —  •        I"  ?f  +  PC  ~f"  Ta 

r*         /)  K-  n 

Cosu=  -=•  =  —  ,  (}f\4\ 

Z  1  ^       ' 


Power  input  is 

E*(^+rf  +  rc  +  ra} 

(165) 


The  copper  loss  and  equivalent  effective  resistance  loss  will  be 

E2 


72  p 

-77r-T7—  — —          —  -y2  (166x 

/  +Xf+Xa 


Electrical  output  is 

E2  S  Xf 


P-PR  = 


/S  Xf  V     / 

f  W  +rf+r€+  raj  +  {Xf  +  xf+  xaj 

The  torque  is 


(167) 


D  =      ~  =  (see  eq.  161)  (168) 

o  H 

VECTOR  DIAGRAM  OF  COMPENSATED  SERIES  MOTORS. 

The  equations  here  given  are  represented  graphically  in  the 
diagrams  of  Figs.  115  and  116,  which  show  the  impedance  (e.m.f. 
for  unit  current)  characteristics  of  the  machines. 

OA=rf 
B  C  =  ra  +  rc 
AB  =  Xf  +  xf 
DC  =  Xa 

D  E  =  —  ^1  at  speed  S 
n 

0  E  =  Z  at  speed  5 

cos  E  0  A  =-  cos  0  =  power  factor  at  speed  5 


SERIES  MOTORS.  -.-  245 

These  characteristics  together  with  the  brush  short  circuiting 
effect  and  other  minor  modifying  influences  will  be  discussed 
later.  It  is  sufficient  here  to  state  that  the  effect  of  the  short 
circuit  by  the  brush  of  a  coil  in  which  an  active  e.m.f.  is  gen- 
erated, both  by  transformer  and  speed  action,  tending  to  in- 
crease the  apparent  impedance  effects  at  high  speeds  is  to  some 
extent  balanced  by  the  fact  that  the  flux  which  causes  the  gen- 
eration of  a  counter  e.m.f.  by  dynamo  speed  action  is  out  of 
phase  and  lagging  with  respect  to  the  line  current  and  that  the 
counter  e.m.f.  therefore,  tends  to  lag  behind  the  current  or  to 
cause  the  current  to  become  leading  with  respect  to  the  counter 
e.m.f.,  so  that  the  neglected  disturbing  influences  tend  to  render 
the  final  effect  quite  small,  the  result  being  that  the  incomplete 
equations  and  corresponding  graphical  diagrams  as  given  above, 
represent  quite  closely  the  observed  performance  characteristics 
of  the  compensated  series  motors. 

INDUCTION  SERIES  MOTOR. 

Excellent  performance  of  the  compensated  alternating-current 
motor  may  be  obtained  by  using  the  field  coil  as  the  load  circuit 
from  the  compensating  coil  employed  as  the  secondary  of  a  trans- 
former, the  armature  being  used  as  the  primary,  as  diagrammat- 
ically  represented  in  Fig.  117.  The  current  which  enters  the 
armature  winding  through  the  brushes  Bl  B2  causes  the  forma- 
tion on  the  armature  core  of  magnetic  poles  having  the  mechan- 
ical direction  of  the  axial  line  joining  the  brushes,  and  the  rate 
of  change  of  the  magnetism  generates  an  electromotive  force  in 
the  compensating  coil.  Due  to  this  electromotive  force,  current 
flows  through  the  locally-closed  circuits  around  the  compensat- 
ing and  field  coils,  and  produces  magnetic  poles  in  the  stationary 
field-cores. 

Consider  now  the  load-circuit  surrounding  the  quadrature 
field-cores.  Since  to  this  winding  there  is  no  opposing  secondary 
circuit,  the  magnetism  in  the  core  will  be  practically  in  time- 
phase  with  the  current  producing  it.  This  current  is  the  sec- 
ondary load-current  of  the  transformer.  As  is  true  in  any  trans- 
former, there  will  flow  in  the  primary  coil  a  current  in  phase 
opposition  to  the  secondary  current  in  addition  to  and  super- 
posed upon  the  primary  no-load  exciting-current.  It  is  thus 
seen  that  the  load-current  in  the  primary  (or  armature)  coil  will 


240 


ALTERNATING  CURRENT  MOTORS. 


be  in  time-phase  opposition  with  the  magnetism  in  the  quadrature 
core.  And,  since  this  current  and  the  magnetism  reverse  signs 
together,  the  torque,  due  to  their  product  and  relative  mechan- 
ical position,  will  remain  always  of  the  same  sign — though 
fluctuating  in  value.  Hence  the  machine  operates  similarly  to  a 
direct-current  series  motor. 


Current 

__&• — i 

O  A 

FIG.  117. — Circuit  and  Vector  Diagrams  of  Induction  Series 
Motor. 

When  the  armature  revolves  at  a  certain  speed,  the  motion 
of  its  conductors  through  the  quadrature  magnetic  field,  gener- 
ates in  the  armature  winding  an  electromotive  force  which  ap- 
pears at  the  brushes  B1  B2  as  a  counter  e.m.f.  This  weakens  the 
effective  electromotive  force  and  therewith  the  armature-current , 
the  armature-core  magnetism,  the  field-current  and  the  field-core 
magnetism.  Thus  there  results  from  increased  speed  of  the  arma- 


SERIES  MOTORS.  247 

ture  a  reduced  torque,  just  as  occurs  in  direct-current  series 
motors.  By  increasing  the  applied  electromotive  force,  an  in- 
crease of  torque  can  be  obtained  even  at  excessively  high  speeds, 
and  the  motor  tends  to  increase  indefinitely  the  speed  of  its  ar- 
mature as  the  applied  electromotive  force  is  increased,  or  as  the 
counter  torque  is  decreased.  There  is  no  tendency  to  attain  a 
definite  limiting  speed  as  is  found  to  be  true  with  revolving 
field  induction-motors  and  repulsion  motors. 

FUNDAMENTAL  EQUATIONS  OF  INDUCTION  SERIES  MOTOR. 

Let  Ea  be  the  counter  transformer  e.m.f.  across  the  armature 
coil,  the  armature  being  stationary. 
Then 

a  see  eq.  (55)          -    -     (169) 


where  /  =  frequency  in  cycles  per  second 
Na  =  effective  number  of  armature  turns 
<j)a  =  maximum  value  of  armature  flux  cutting  the 
compensating  coil 

Na  =  £-  see  eq.  (56)  (170) 

Z  n 

where  C  is  the  actual  number  of  conductors  on  the  armature,  a 
bipolar  model  being  assumed. 

7          f  C  <f>g 


Let  Nc  =  effective  number  of  turns  on  the  compensating  coil, 
then 


_  e>a      _  EaNc 

\/2~108  Na 

where  Ec  =  transformer  e.m.f.  of  the  compensating  coil. 
Let  Nf  =  effective  number  of  turns  on  the  field  coil 
then 


__ 

-*  (173) 


248  ALTERNATING  CURRENT  MOTORS. 

where  Ef  —  impressed  e.m.f.  of  the  field  coil 

<j>f  —  maximum  value  of  field  flux 

Ef  =  Ec,  hence  Nc  <j>a  =  Nf  </>/  (174) 

and 

&  -       c 


Let  Ev  be  the  e.m.f.  counter  generated  at  the  brushes  B^  B2 
(Fig.  117)  by  speed  action  due  to  the  cutting  of  the  flux  <^;  by  the 
armature  conductors  C  at  speed  V  revolutions  per  second,  then 


(59)  (176) 


V  =  S  /  (177) 

where  S  is  the  speed  with  synchronism  as  unity. 
Combining  (171),  (176)  and  (177) 


<»«> 


Let  n  be  the  ratio  of  effective  field  to  compensating  coil  turns. 

Ev  = a     see  eq.  (123)  (179) 

This  electromotive  force  is  in  time -phase  with  the  field  flux 
<£/,  is  in  phase  opposition  with  the  line  current  and  hence  is  in 
time  quadrature  (leading)  with  respect  to  the  e.m.f.  Ea.  The 
impressed  electromotive  force  E  is  balanced  by  the  two  com- 
ponents, Ev  and  Ea  so  that 

E  =  VEJ+E*  (iso) 

5+1  (181) 

On  the  basis  of  unit  line  current,  the  electromotive  forces  may 
be  treated  as  impedances,  as  was  done  with  the  repulsion-series 
and  compensated-series  motors. 


Z  =  X,  Jj£-  +  i  (182) 


SERIES  MOTORS.  249 

where 

Xt  -X 


Xt  being  the  combined  reactance  effect  of  the  field,  compensating 
coil  and  armature  circuits. 
The  power  factor  is, 

_s; 

R  n  S 

(183) 


<  -  & 

which  when  S  =  1  or  at  synchronism,  reduces  to 

cos  0  =      .  l  (184) 

Vl+n2 

the  interpretation  of  which  is  that  the  power  factor  at  synchro- 
nism can  be  caused  to  approach  unity  quite  closely  by  the  use  of 
a  small  value  of  n,  that  is,  by  employing  a  small  ratio  of  field  to 
compensating  coil  turns.  With  increase  of  speed  the  power  fac- 
tor continually  increases  for  any  value  of  n. 
The  line  current  is 

7_E_==E_         1  En 


Z      Xt     I  & Xt  VSM^2  (185) 

\  n2"  "^ 
The  power  is 

S_ 
E2         n  E2  S  n 


•>         '    I  "       •*" 

n2 

which  becomes  negative  when  5  reverses,  or  the  machine  oper- 
ates as  a  generator  when  driven  against  its  natural  tendency  to 
rotation. 

The  torque  is 

P_  E2n  _P_X1 

~  S     ~  Xt(S2+n2}  ~      n 

which  is  max;mum  at  maximum  current  and  retains  its  sign 
when  5  is  reversed. 


250  ALTERNATING  CURRENT  MOTORS. 

At  starting,  the  torque  is 

D«=t'£   ~:     ;         f        (188) 
at  synchronous  speed,  the  torque  is 


D>  - 1  •  oiV)  <189) 


and 


which  when  n  =  1  reduces  to 

D*  _      1        .5 

5.  -  IT!  • 

and  can  be  given  any  desired  value  by  a  proper  selection  of  n, 
see  eq.  (153).  A  relatively  low  value  of  n  would  produce  a 
machine  having  the  torque  characteristics  of  the  direct  current 
series  motor  and  hence  one  suitable  for  traction.  See  eq.  (162). 

It  remains  to  investigate  the  relation  of  the  currents  in  the 
compensating  coil  and  in  the  armature  circuit  (the  secondary  and 
primary  of  the  assumed  transformer). 

Let  ia  be  the  current  which  would  flow  in  the  armature  when 
the  field  coil  circuit  is  open.  Then  ia  is  the  exciting  current  of 
the  assumed  transformer  and  it  has  a  value  such  that  its  product 
with  the  effective  number  of  armature  turns,  forces  the  flux,#Q, 
demanded  by  the  impressed  e.m.f.,  through  the  reluctance  of 
their  paths  in  the  magnetic  structure,  in  line  with  the  brushes 
Bl  B2  (Fig.  117).  When  the  field  circuit  is  closed  there  flows 
through  the  field  and  compensating  coil  a  current  if,  of  a  value 
such  that  its  magnetomotive  force  when  flowing  through  the 
field  turns  Nf,  produces  the  flux  4>f  demanded  by  the  e.m.f.  Ef  or 
Ec.  The  current  if  is  in  time-phase  with  the  flux  4>f  and  hence 
is  in  time  quadrature  with  the  e.m.f.  Ec.  The  current  ia  is  in 
phase  with  the  flux  tf>a  and  in  time  quadrature  with  Ea  or  Ec. 
When  the  field  circuit  is  closed  a  current  equal  in  magnetomotive 
force  and  opposite  in  phase  to  if  is  superposed  upon  ia  in  the 
primary  (armature)  circuit.  These  two  currents  are  directly  in 
phase  so  that  the  resultant  current  becomes 

I=ia  +  pif  (192) 


SERIES  MOTORS.  251 

where  p  is  a  proportionality  constant  the  value  of  which  will  be 
discussed  later. 

Since  both  ia  and  if  reach  their  maximum  values  simultane- 
ously with  (j)f,  one  is  led  to  the  highly  interesting  conclusion  that 
even  the  exciting  current  ia  is  effective  in  producing  torque  by 
its  direct  product  with  the  field  magnetism,  and  that  under 
speed  conditions  both  ia  and  p  if  are  equally  effective  (per 
ampere)  in  producing  power. 

The  relative  values  of  ia  and  if  and  of  p  may  be  approximated 
as  follows: 

Assuming  similar  conditions  for  the  three  coils,  the  field,  the 
compensating  and  the  armature  circuits,  —  equal  reluctance  — 

=*1"1  (193) 


Nf  =  n  Nc  (194) 

Nc  fa  =  -Nf  <f>f    see  eq.  (174)  (195) 


(197) 
Nf  (j>a   "          Mf  Ncn" 

From  transformer  relations  there  is  obtained  the  equation 

^  =  p     see  eq.  (192)  (198) 

Combining  (197)  and  (198) 

if  = 

pn 


*  (199) 


Combining  (199)  and  (192) 

/==^(1+^)  (200) 

Comparing  (199)  and  (200), 


252  ALTERNATING  CURRENT  MOTORS. 

CORRECTIONS  FOR  RESISTANCE  AND  LOCAL  LEAKAGE  REACTANCE. 
The  relations  above  expressed  depend  upon  certain  assump- 
tions as  to  the  reluctance  in  line  with  the  armature  circuit  and 
the  field  coil,  and  will  be  modified  if  the  assumptions  made  are 
not  applicable  to  the  motor  as  constructed.  As  a  method  of 
reviewing  the  problem,  in  a  general  way,  however,  the  assump- 
tion made  and  the  conclusions  drawn  therefrom  are  sufficiently 
exact.  In  the  determination  of  the  equations  used  above,  an 
ideal  motor  has  been  considered,  the  resistance  and  local  leakage 
reactance  effects  being  neglected.  Actual  operating  conditions 
may  be  more  closely  represented  as  follows: 

Let 

rf  =  resistance  of  field  coil. 

rc  =  resistance  of  compensating  coil. 

ra  =  resistance  of  armature. 

xf  =  local  leakage  reactance  of  field  coil. 

xc  =  local  leakage  reactance  of  compensating  coil. 

Xf  =  local  leakage  reactance  of  armature  circuit. 

Then  the  copper  loss  of  the  motor  circuits  will  be 

P  Rm  =  Pra  +  f,  (n  +  rc'  =  P  [r0  +  (~^p]       (202) 

where  Rm  is  the  effective  equivalent  value  of  the  motor-circuit 
resistance,  that  is, 

(203> 


Similarly  it  may  be  shown  that  the  equivalent  effective  value 
of  the  local  leakage  reactance  of  the  motor-circuit  is 


(204) 


Combining  equations   (182),    (203)   and    (204),  the  apparent 
impedance  of  the  motor-circuits  becomes 


Xt, 

+r°+(p^)~         laffl(^?+p)i 

(205) 


SERIES  MOTORS.  253 

The  power  factor  is 

5  Xt  rt  +  rc 

R  n          r    a         (-hn2  4-  <h\2 

COS  0=7?-  = 


(206) 
The  current  is  •=-  =  I.  (207) 

JL* 

The  input  =  E  I  cos  0.  (208) 

The  output  is  P  =  £  7  cos  0  -  72  #w.  (209) 


P  = 


(211) 

Z/'  1H  '«• 

The  torque  is 

D  =  f-  4-  — —  see  eq.  (187)  and  eq.  (168)  (212) 

wj  /yi> 

VECTOR  DIAGRAM  OF  INDUCTION  SERIES  MOTOR. 

The  graphical  diagram  of  Fig.  117  represents  the  above  im- 
pedance equations  (e.m.f.  for  unit  current),  where 

(213) 


AD  -X* +*«+-*  (214) 


D  F  =  — ~  at  speed  5  (215) 

O  F  =  Z  at  speed  S  (216) 

cos  F  O  A  =  cos  6  =  power  factor  at  speed  S        (217) 


254  ALTERNATING  CURRENT  MOTORS. 

Although  neglecting  certain  modifying  effects,  the  graphical 
diagram  represents  quite  closely  the  observed  performance  char- 
acteristics of  the  induction-series  motor.  An  inspection  of  equa- 
tion (205)  will  show  that  certain  values  there  given  may  be 
represented  by  others  of  much  simplified  nature  since  various 
terms  there  contained  are  -constant  in  any  chosen  motor. 

Let,  therefore, 

'  -"-    -  •  (218) 


P  =  —  (220) 

then  the  apparent  impedance  becomes, 

Z  =  V(R  +  PS)2  +  X*  (221) 

the  power  factor  is, 

cos  0  =     ,     B  +  PS   =  (222) 

V(R+PS) 


which  continually  approaches  unity  with  increase  of  speed. 

GENERATOR  ACTION  OF  INDUCTION  SERIES  MOTOR. 

Let  rotation  of  the  armature  in  the  direction  produced  by  the 
electrical  (its  own)  torque  be  considered  positive.  Then  may 
rotation  in  the  contrary  direction  (against  its  own  torque)  be 
considered  negative.  Since  the  power  component  of  the  motor 
impedance  has  a  certain  value  at  zero  speed,  and  increases  with 
increase  of  speed,  it  should  follow  that  by  driving  the  rotor  in  a 
negative  direction  the  apparent  power  component  will  reduce 
to  zero  and  disappear.  The  power  factor  then  reduces  to  zero 
and  the  current  supplied  to  the  motor  will  represent  no  energy 
flowing  either  to  or  from  the  motor. 

This  will  be  apparent  from  the  relations  above  set  forth,  as 
well  as  by  the  relations  algebraically  expressed  by  the  equation 


the  negative  sign  being  due  to  the  direction  of  rotation  and  the 
expression  reducing  to  zero  for  zero  value  of  the  apparent  power 


SERIES  MOTORS. 


255 


component,  R  —  P  S.  A  further  increase  of  speed  in  the  nega- 
tive direction  will  cause  the  expression  for  the  power-factor  and 
for  the  power,  to  become  negative,  the  interpretation  of  which  is 
that  the  machine  is  now  being  operated  as  a  generator  and 
hence  is  supplying  energy  to  the  line,  that  is,  energy  is  flowing 
from  the  machine.  Fig.  118  which  gives  the  observed  perform- 
ance characteristics  of  a  certain  induction-series  motor,  will 
serve  to  show  to  what  extent  these  theoretical  deductions  may 
be  realized  in  an  actual  machine.  If,  then,  during  operation  as 


-100 


-16        -12 


12         .16 


20         24 


-8-4048 
Speed  in  100  R.P.M. 

FIG.  118. — Test  Characteristics  of  Induction-Series  Motor. 

a  motor  at  a  certain  speed,  the  quadrature  field  flux  be  relatively 
reversed  with  reference  to  the  brush  axial-line  field  flux,  so  as  to 
tend  to  drive  the  armature  in  the  opposite  direction,  not  only 
will  a  braking  effect  be  produced  by  such  change  but  energy 
will  be  transmitted  from  the  machine  to  the  line. 

BRUSH  SHORT-CIRCUITING  EFFECT. 

The  effect  of  the  short  circuit  by  the  brush  of  a  coil  in  which 
an  active  e.m.f.  is  generated,  which  has  been  omitted  in  the 


256  ALTERNATING  C'JRREVT  MOTORS. 

above  equations,  though  completely  included  in  the  test  curves, 
may  be  treated  as  follows.  Referring  to  Fig.  117,  it  will  be  seen 
that  at  any  speed  5  there  will  be  generated  in  the  coil  under  the 
brush  by  dynamo  speed  action  an  e.m.f. 

es  =  K  <t>a  S     see  eq.  (43)  (224) 

where  K  is  constant.  This  e.m.f.  is  in  time-phase  with  the  flux 
</)a.  In  this  coil  there  will  also  be  generated  an  e.m.f.,  ef,  by 
the  transformer  action  of  the  field  flux,  such  that 

ef  =  K  <j>f     see  eq.  (44)  (225) 

This  e.m.f.  is  in  time  quadrature  to  <f>f.  Since  </>/  and  <f>a  are  in 
time  phase,  the  component  e.m.f. 's  acting  in  the  coil  under  the 
brush  are  in  time  quadrature,  so  that  the  resultant  e.m.f.  is 

(226) 
^  =  n     see  eq.  (175)  (227) 

(228) 


combining  equations  (169)  and  (181) 

ZxfN.f. 


.. 
V2-  10"  -Vi?-1  (229) 

\/2  .  108  .  E 

=  (230) 


combining  (230)  and  (228) 

K  VT  .   103  E 
b  =          2xfNa 


^  +1 


(231) 


S2  >«2  +  1 


where  A  is  a  constant  as  found  above. 

When  n  =  1,  Eb  is  constant,  independent  of  the  speed,  while 
when  n  is  very  small  Eb  is  large  at  zero  speed  and  continually 


SERIES  MOTORS.  257 

decreases  with  increase  of  speed.     When  S  =  1  or  at  synchronous 
speed 

-      KvTip  (233) 

quite  independent  of  the  value  of  n. 

The  relative  impedance  effect  on  E&  can  be  determined  by 
combining  equations  (232)  and  (185)  thus 


Eb       A  xt         S2  n2  +  1          ,  _ 

7   -  £¥    -s^    •  ^52+"2  (234) 


2+l  (235) 


B  being  a  constant.  The  interpretation  of  equation  (235)  is 
that  the  apparent  impedance  effect  of  the  short  circuit  by  the 
brush  consists  of  two  components  in  quadrature,  one  component 
being  of  constant  value  and  the  other  varying  directly  with  the 
speed.  Experimental  observations  fully  confirm  these  theoret- 
ical conclusions,  and  show  that  the  increase  in  apparent  reactive 
effect  with  increase  of  speed  for  motor  operation  is  approxi- 
mately counterbalanced  by  the  lagging  counter  e.m.f.  (leading 
current)  effect  of  the  time-phase  displacement  between  exciting 
current  and  field  magnetism  as  has  been  mentioned  previously 
and  as  will  be  dwelt  upon  subsequently.  During  generator 
operation,  that  is,  with  negative  value  of  5,  the  apparent  re- 
active effect  of  the  short  circuit  at  the  brush  adds  directly  to 
the  lagging  field  flux,  counter  e.m.f.  effect  and  therefore,  the 
apparent  reactance  of  the  motor  circuits  increases  rapidly  with 
increase  of  speed  in  the  negative  direction,  though  remaining 
practically  constant  for  all  values  of  positive  speed.  These  facts 
will  be  appreciated  from  a  study  of  the  test  characteristics  of 
the  induction  series  machine  throughout  both  its  generator  and 
motor  operating  range  as  shown  in  Fig.  118. 

HYSTERETIC  ANGLE  OF  TIME-PHASE  DISPLACEMENT. 

Mention  has  frequently  been  made  of  the  fact  that  in  the 
development  of  the  equations  for  expressing  the  performance  of 
the  various  types  of  series  motors  the  effect  of  the  hysteretic 
angle  of  time  -phase  displacement,  between  the  magnetizing  force 


258  ALTERNATING  CURRENT  MOTORS. 

and  the  magnetism  produced  thereby  has  been  neglected.  In 
a  closed  magnet  path  operated  at  a  density  below  saturation  the 
tangent  of  the  angle  of  time-phase  displacement  will  be  approxi- 
mately unity  —  depending  for  its  exact  value  upon  the  quality 
of  the  magnetic  material.  Consider  the  magnetic  and  electric 
circuits  of  the  machine  treated  as  a  stationary  transformer. 
The  hysteresis  loss  will  be,  in  watts, 

/B-'-'         .^-'       (236) 


where  A  =  cross  sectional  area  of  magnetic  path 

/  =  length  of  magnetic  path  (in  centimeters) 
Bm  =  maximum  magnetic  density  (c.g.s.) 

The  electromotive  force  counter  generated  in  the  transformer 
.coil  having  N  turns  will  be,  in  effective  volts, 

2nfABmN 

~Wi#~ 

The  current  to  supply  the  hysteresis  loss  will  be 

Wk     .0021  M  *  g-'-*        V2Vf        nm4r9,R.6 
Ih=~E-   '-  2xfABmN~    •        Iff-  mlBm 

With  a  permeability  of  ju  the  magnetizing  component  of  tne 
no-load  current  will  be 

A  Em  I  _10_      BnJ, 

^    =  4  TT       .      ._  .  ,  "  4  vT  *  '   fi  N 

—  ftA^/^N  (239) 

For  a  certain  value  of  permeability,  depending  upon  the  mag- 
netic density,  the  hysteresis  current  and  the  magnetizing  cur- 
rent become  equal  in  value.  Thus  when  the  two  components 
of  the  no-load  exciting  current  become  equal  I  /*  =  Ih, 

.0021fBm-«  Bm.l.W 

V2  2xN          ~  4  n  V2.f-N 

from  which  is  obtained, 

/<=119£m-<  (250) 

The  meaning  of  equation  (250)  is  that  with  a  permeability  of 
the  value  there  designated,  the  hysteresis  current  and  the  no-load 


SERIES  MOTORS.  259 

exciting  current  are  equal  in  value  and  that  the  resultant  current 
v7/t2  +  /  ^2  is  displaced  from  the  flux  by  a  time-phase  angle  whose 
tangent  (equal  at  all  times  to  the  ratio  of  /  JJL  to  Ih)  is  unity,  as 
stated  previously.  For  commercial  laminated  steel  operated  at 
densities  below  saturation,  the  permeability  differs  but  slightly 
from  the  value  given  by  the  equation  (250),  though  with  increase 
of  magnetic  density  above  7,000  lines  per  square  centimeter  the 
permeability  falls  off  rapidly  and  the  tangent  of  the  angle  of 
displacement  between  flux  and  current  becomes  correspondingly 
increased. 

In  an  open  magnetic  circuit  the  permeability  of  a  portion  of 
the  path  reduces  from  the  value  approximately  represented  by 
the  equation  (250)  to  a  value  of  unity,  producing  a  very  marked 
effect  upon  the  hysteretic  angle  of  displacement  between  flux 
and  current. 

Let  /  =  length  of  path  in  magnetic  material  of  permeability  /*, 
d  =  length  of  path  in  air, 

then,  assuming  that  permeability  is  as  represented  by  equation 
(250),  the  tangent  of  the  angle  of  time-phase  displacement  be- 
tween flux  and  magnetizing  force  is  such  that 


(251) 


the  significance  of  which  equation  is  that  the  flux  lags  behind 
the  current  producing  it,  by  an  angle  which  depends  for  its  value 
largely  upon  the  ratio  of  the  air-gap  to  the  length  of  the  mag- 
netic path.  Assigning  values  to  /*,  /  and  d,  it  will  be  seen  that 
in  any  practical  case  the  angle  d  must  be  quite  small, — seldom 
more  than  2  degrees. 

It  should  be  carefully  noted  that  a  slight  error  is  introduced 
on  account  of  the  fact  that  the  permeability  of  commercial  mag- 
netic material  undergoes  a  cyclic  change  with  each  alternation 
of  the  current,  and  that,  independent  of  the  angle  of  time-phase 
displacement  between  flux  and  current,  the  shape  of  the  waves 
representing  the  time-values  of  the  two  can  not  both  be  sinu- 
soidal, and  that  in  assigning  a  value  to  the  angle  of  time-phase 
displacement  between  the  flux  and  current,  the  lack  of  similarity 
of  the  two  waves  has  been  neglected. 


260  ALTERNATING  CURRENT  MOTORS. 

POWER  FACTOR  OF  COMMUTATOR  MOTORS. 

Under  speed  conditions  the  e.m.f.  counter  generated  by  the 
cutting  of  the  armature  conductors  across  the  field  magnetism, 
varies  in  value  with  the  magnetism,  and  hence  it  must  have  a 
wave  shape  of  time-value  similar  in  all  respects  to  that  of  the 
field  flux,  and  must  have  a  time-phase  position  with  reference 
to  the  field  current  quite  the  same  as  that  of  the  magnetism. 
The  counter  generated  speed  e.m.f.  must,  therefore,  lag  behind 
the  current  by  an  angle  whose  tangent  is  as  given  by  equation 
(251).  Now  since  the  counter  e.m.f.  lags  behind  the  current, 
the  current  must  lead  the  counter  e.m.f.  by  the  same  angle — 
a  fact  which  has  been  mentioned  previously. 

With  motors  having  air  gaps  of  sizes  demanded  by  mechanical 
clearance,  the  inherent  angle  of  lead  is  quite  small,  and  its  effect 
upon  the  power  factor  is  neutralized  by  the  effect  of  the  short 
circuit  by  the  brush  of  a  coil  in  which  is  generated  an  e.m.f.  by 
both  transformer  and  speed  action  when  the  machine  is  operated 
as  a  motor.  When  the  machine  is  operated  as  a  generator,  how- 
ever, the  hysteretic  angle  and  the  angle  due  to  the  short  circuit- 
ing effect  are  in  a  direction  such  as  to  be  additive  to  the  station- 
ary reactive  effect  of  the  motor  circuits  and,  therefore,  during 
generator  operation  the  power  factor  is  lower  than  during  motor 
operation,  as  shown  in  Fig.  118. 

While  the  angle  of  lead  due  to  the  hysteretic  effect,  even  when 
the  machine  is  running  as  a  motor,  is  in  any  case  quite  small  and 
its  good  effects  cannot  be  availed  of,  it  is  possible  by  means  of 
certain  auxiliary  circuits  to  give  to  the  angle  of  time-phase  dis- 
placement between  the  line  current  and  the  flux  any  value  de- 
sired, and  thus  to  cause  the  operating  power  factor  to  become 
unity  or  to  decrease  with  leading  wattless  current,  as  is  shown 
below. 

RESISTANCE  IN  SHUNT  WITH  FIELD  WINDING. 

Fig.  119  represents  diagrammatically  the  circuits  of  a  con- 
ductively  compensated-series  motor  in  parallel  with  the  field 
coil  of  which  is  placed  a  non-inductive  resistance.  Consider 
first,  ideal  conditions  in  which  the  armature  and  compensating 
coils  are  without  resistance  and  the  compensation  is  complete 
so  that  these  two  circuits,  treated  as  one,  are  without  inductance. 
The  field  coil  is  without  resistance  but  constitutes  the  reactive 
portion  of  the  motor  circuits. 


SERIES  MOTORS. 


261 


When  the  armature  is  stationary  the  circuit  through  the  re- 
sistance being  open,  the  current  taken  by  the  machine  has  a 
value  determined  by  the* ratio. of  the  impressed  e.m.f.  and  the 
reactance  of  the  field  coil.  This  current  lags  90  time  degrees 


}  Compensaltng  Cti&L 


Es=  Speed  E.M.F. 


If  =  Field  Current 

FIG.  119. — Circiut  and  Vector  Diagrams  of  Compensated 
Series  Motor  with  Shunted  Field  Coil. 

behind  the  e.m.f.  across  the  field  coils.  When  a  resistance  is 
placed  in  shunt  to  the  field  coil,  current  flows  therethrough, 
quite  independently  of  the  field  current.  The  current  taken 
by  the  resistance  is  in  time-phase  with  the  e.m.f.  impressed 
upon  the  field  coil. 


262  ALTERNATING  CURRENT  MOTORS. 

In  Fig.  119  let  O  I  =  If  represent  the  field  current,  assumed 
always  of  unit  value.  O  D  =  Ef  is  the  e.m.f.  impressed  across 
the  field  coil  and  the  shunted  resistance.  Ir  is  the  current  taken 
by  the  resistance.  O  C  =  /,  the  current  which  flows  through 
the  armature  and  compensating  coil  or  the  resultant  current 
taken  by  the  motor  has  a  value  represented  by  the  equation 

i  =  V/TTT2  (252> 

and  has  a  phase  displacement  /?  with  reference  to  the  field  cur- 
rent such  that 

O        lr 

tan/?=]7  (253) 

With  unit  value  of  field  current,  under  speed  conditions,  the 
e.m.f.,  Es,  (D  F  of  Fig.  119)  counter  generated  at  the  brushes, 
due  to  the  presence  of  the  field  flux,  will  be  proportional  directly 
to  the  speed  and  in  time-phase  with  the  field  current.  Thus 
this  component  of  the  counter  e.m.f.  of  the  motor  is  in  no  wise 
affected  by  the  presence  of  the  current  through  the  shunted 
resistance.  At  a  certain  speed,  the  counter  generated  armature 
em.f.  will  have  a  value  represented  by  the  line  D  F  Fig.  119 
the  resultant  e.m.f.  E  =  O  F  being  the  vector  (quadrature) 
sum  of  the  speed  e.m.f.  and  the  stationary  e.m.f.  Es,  that  is 

E  =  VEf  +  E?  (254) 

and  has  a  time-phase  a  position  with  reference  to  the  speed  e.m.f. 
Es  such  that 

tana  =  §7  .(255) 

An  inspection  of  Fig.  119  will  show  that  under  operating  con- 
ditions, the  angle  of  time-phase  displacement  between  the  cur- 
rent and  the  electromotive  force,  6,  has  a  value  represented  by 
the  equation 

6  =p-a  (256) 

or  the  current  leads  the  e.m.f.  by  the  angle  0.  At  a  certain 
critical  speed  for  each  value  of  shunted  resistance,  or  at  a  certain 
value  of  resistance  for  any  given  speed,  the  angle  0  reduces  to 
zero,  and  the  power  factor  of  the  motor  becomes  unity. 

It  is  interesting  to  observe  the  effect  of  removing  the  resist- 
ance from  in  shunt  with  the  field  circuit.  Since  the  current 


SERIES  MOTORS.  263 

taken  by  the  resistance  is  90  time-degrees  from  the  field  flux, 
the  resultant  torque  due  to  the  product  of  this  component  of 
the  current  and  the  flux  is  of  zero  value,  the  instantaneous 
torque  alternating  at  double  the  circuit  frequency.  The  cur- 
rent through  the  resistance,  therefore,  contributes  in  no  way 
to  the  power  of  the  machine  or  to  the  counter- generated,  arma- 
ture-speed e.m.f.,  and  when  the  circuit  through  the  resistance 
is  opened  no  effect  whatsoever  is  produced  upon  the  value  of 
the  current  taken  by  the  field  coil,  the  counter  e.m.f.  or  the  torque 
of  the  machine.  It  is  apparent,  therefore,  that  the  use  of  the 
shunted  resistance  increases  the  circuit  current  in  a  certain 
definite  proportion,  the  added  component  being  a  leading 
"  wattless  "  current  under  speed  conditions.  If  a  reactance  be 
placed  in  parallel  with  the  field  coil,  the  current  which  flows 
therethrough  will  be  in  time-phase  with  the  field  flux,  and  the 
torque  produced  thereby  will  add  to  the  torque  due  to  the  field 
current  and  it  will  affect  directly  the  whole  performance  of  the 
machine.  The  current  taken  by  a  condensance  in  shunt  with 
the  field  coil  will  be  in  time-phase  opposition  to  the  field  current 
and  will  tend  to  decrease  directly  both  the  circuit  current  and 
the  armature  torque.  An  excess  of  condensance  will  cause  the 
torque  to  reverse  and  the  machine  to  act  as  a  generator  even 
when  the  speed  is  in  a  positive  direction.  When  the  condensance 
and  the  field  reactance  are  just  equal,  the  circuit  current  re- 
duces to  zero  and  the  torque  disappears.  Under  the  conditions 
here  assumed,  the  counter  generated  e.m.f.  at  the  armature  re- 
mains proportional  to  the  product  of  the  field  flux  and  the  speed, 
and  there  appears  the  remarkable  combination  of  zero  current 
being  transmitted  over  a  certain  counter  e.m.f.  (that  is,  through 
infinite  impedance)  to  divide  into  definite  active  currents  at 
the  end  of  the  transmission  circuits. 

Loss  DUE  TO  USE  OF  SHUNTED  RESISTANCE. 

From  what  has  been  demonstrated  above,  it  is  seen  that 
shunted  condensance  acts  to  take  current  in  phase  opposition 
and  to  decrease  the  torque;  reactance  takes  current  directly  in 
phase,  and  increases  the  torque,  while  resistance  takes  current 
in  leading  quadratures  with  the  field  current  and  has  no  effect 
upon  the  torque.  It  is  evident  that  the  improvement  in  power 
factor  due  to  the  use  of  the  resistance  is  advantageous  provided 


264 


ALTERNATING  CURRENT  MOTORS. 


the  losses  caused  by  the  resistance  are  not  excessive.  Referring 
to  Fig.  119,  when  the  resistance  is  not  used  the  power  taken  by 
the  machine  under  speed  conditions  is 


P  =  OI.OF.cosFOI  =  If  E  cos  a  =  //  Es 


(257) 


When  the  machine  is  stationary,  the  power  absorbed  by  the 
resistance  is 

Pr  =  CI.OD  =  IrEf  (258) 

When  the  motor  is  running  with  shunted  field  coil,  the  power 
delivered  to  the  machine  is 


Pt  =  OC.OF.cosCOF  =/£cos0 


(259) 


Current  2-3  4  5  G 

Ohms-Volts  at  One  Amp. 

FIG.  120. — Observed  e.m.f . — Current  Characteristics  of  Plain 
Series  Motor  with  Shunted  Field  Coils. 


6  =  ft  -  a  (260) 

cos  6  =  cos  ft  cos  a  +  sin  ft  sin  a  (261  ) 

Pt  =  /cos/?.  E  cos  a  +7  sin/?.  E  sin  a  (262) 

Pt  =  IfEs  +  IrEj  =  P  +  Pr  (263) 


The  significance  of  equation  (263)  is  that  the  power  absorbed 
is  that  incident  to  the  use  of  the  resistance,  and  that  for  a  given 
current  it  is  unaffected  by  the  speed  e.m.f.  Thus  the  current 
taken  by  the  resistance  multiplies  into  the  stationary  trans- 
former e.m.f.  to  give  the  actual  watts  absorbed  while  the  same 


SERIES  MOTORS.  265 

current  multiplies  into  the  speed  e.m.f.  to  give  apparent  leading 
wattless  power. 

In  the  derivation  of  the  above  equations  ideal  conditions  have 
been  assumed,  which  cannot  be  obtained  in  a  practical  motor. 
Fig.  120  represents  the  observed  e.m.f. -current  characteristics 
of  a  certain  plain,  uniform  reluctance  motor  (see  Fig.  114) 
with  shunted  field  coils,  and  serves  to  show  that  even  such  an 
unfavorable  machine  may  be  caused  to  operate  at  unity  power 
factor  at  any  speed  greater  than  about  one-half  synchronism. 


CHAPTER  XVI. 

PREVENTION  OF  SPARKING  IN  SINGLE-PHASE  COMMUTATOR 

MOTORS. 

TRANSFORMER  ACTION  WITH  STATIONARY  ROTOR. 

The  greatest  difficulty  which  has  been  encountered  in  the 
design  of  alternating-current  motors  of  the  commutator  type 
has  resided  in  the  unavoidable  e.m.f.  produced  at  the  coil  under 
the  brush  due  to  the  variation  in  the  field  magnetism.  When 
the  armature  is  stationary,  there  exists  an  appreciable  electro- 
motive force  between  the  terminals  of  each  coil,  the  field  coils 
acting  as  the  primary  and  the  armature  coils  in  the  neighbor- 
hood of  the  brushes  as  the  secondary  of  a  transformer,  as  indi- 
cated diagrammatically  in  Figs.  121  and  122. 

In  motors  of  the  repulsion  type  or  by  special  magnetizing 
coils  placed  on  any  of  the  other  types  of  motors,  it  is  possible 
to  neutralize  the  transformer  e.m.f.  in  the  coil  under  the  brush 
by  a  speed  generated  e.m.f.  when  the  rotor  is  in  motion.  See 
equations  (48)  and  (101).  The  neutralization  of  the  transformer 
e.m.f.  under  starting  conditions  is  not  wholly  impossible,  but 
it  may  be  stated  that  such  neutralization  involves  certain  com- 
plications which  are  not  desirable  in  a  commercial  motor. 

When  the  rotor  is  at  rest  the  full  effect  of  the  transformer 
e.m.f.  is  felt  at  the  brushes,  quite  independent  of  the  type  of 
motor  employed  and  it  may  fairly  be  said  that  all  simple  forms 
of  alternating-current  commutator  motors  are  equally  dis- 
advantageous with  regard  to  the  sparking  at  starting.  The 
current  which  flows  through  the  short-circuit  coil  by  way  of 
the  brush  is  ordinarily  of  large  value,  and  it  produces  an  exces- 
sive heating  of  the  brush,  the  commutator  segments  and  the 
coil.  Moreover,  the  rupture  of  this  current  when  the  brush 
passes  from  one  commutator  segment  to  the  next  produces 
destructive  arcing  at  the  brushes,  and  its  presence  is  in  general 
detrimental  to  the  perfect  performance  of  the  machine.  To 
the  evil  effects  of  this  local  current  in  the  short-circuited  coils 

266 


SPARKING  IN  COMMUTATOR  MOTORS.  267 

may  be  attributed  the  slow  progress  which  had  been  made  in 
the  development  of  the  commutator  type  of  alternating  current 
machines  previous  to  the  last  few  years. 

INTERLACED  ARMATURE  WINDINGS. 

The  short  circuiting  effect  may  be  largely  eliminated  by  using 
two  or  more  interlaced  armature  windings,  so  arranged  that 
the  brush  cannot  span  the  commutation  sufficiently  to  connect 
two  bars  of  the  same  winding.  As  usually  applied,  this  method 
is  not  satisfactory  on  account  of  the  fact  that  the  current  in 
each  winding  must  be  completely  interrupted  whenever  the 
corresponding  bar  passes  from  contact  with  the  brush.  The 
interruptions  occur  at  a  frequency  depending  upon  the  speed 
of  the  rotor  and  the  number  of  commutator  segments,  and  they 
result  in  serious  sparking  and  pitting  at  the  commutator  equally 
as  disadvantageous  as  that  caused  by  the  short-circuiting. 
That  is  to  say,  the  starting  conditions  have  been  slightly  im- 
proved but  the  running  conditions  have  become  much  worse. 

It  has  also  been  proposed  to  divide  the  armature  circuits 
into  three  distinct  interlaced  windings,  the  current  being  led 
into  the  armature  by  way  of  two  separately  insulated  brushes 
at  each  neutral  point.  By  connecting  the  brushes  in  pairs  to 
the  terminals  of  two  distinct  secondaries  of  a  single  transformer, 
the  current  for  the  different  armature  windings  shifts  from  one 
secondary  coil  to  the  other;  but  at  no  time  is  any  armature  or 
transformer  circuit  broken.  The  method  here  outlined  renders 
the  short-circuiting  effect  a  minimum,  and  it  possesses  consider- 
able merit  in  this  respect.  However  the  method  has  not  been  ap- 
plied extensively  in  commercial  practice,  probably,  on  account  of 
the  involved  electrical  and  mechanical  complications. 

USE  OF  SERIES  RESISTANCE. 

Of  the  many  methods  which  have  been  proposed  for  mini- 
mizing the  effect  of  the  short-circuited  e.m.f.  in  the  coil  under 
commutation,  those  which  involve  the  use  of  resistances  in 
series  with  the  coil  have  proven  to  be  the  most  successful. 
Fig.  121  shows  the  method  by  which  the  resistances  are  inserted 
in  circuit  with  the  coil  under  the  brush ;  the  armature  winding 
is  closed  on  itself  and  is  connected  to  the  commutator  through 
resistance  leads.  These  leads  serve  the  same  function  as  the 


ALTERNATING  CURRENT  MOTORS. 


preventive  coils  used  in  alternating-current  work  when  passing 
from  one  tap  to  another  of  a  transformer.  In  fact  this  armature, 
in  one  sense,  may  be  considered  as  a  transformer  with  a  lead 
brought  out  from  each  coil  through  a  resistance  to  a  contact 
piece,  the  various  contact  pieces  being  assembled  together  to 
form  a  commutator,  as  shown  diagrammatically  in  Fig.  121. 
The  function  of  the  "  preventive  "  resistance  leads  is  to  re- 
duce the  short-circuit  current,  when  passing  from  one  bar  to 
to  the  next  to  a  desirable  low  value.  As  far  as  concerns  com- 
mutation it  is  desirable  that  these  resistances  be  as  large  as 
possible,  while  the  loss  of  power  due  to  the  passage  of  the  main 
motor  current  through  them  dictates  that  their  value  be  kept 

Armature  Lead 
Commutator 


FIG.  121. — Internal  Preventive  Resistance  for  Commutator  Motor. 


quite  small.  It  is  evident,  therefore  that  there  is  some 
intermediate  condition  which  gives  the  most  efficient  results, 
both  as  regards  the  economy  of  power  and  the  commutation 
of  the  current. 

POWER  LOST  IN  RESISTANCE  LEADS. 

It  is  worthy  of  note  that  although  the  prime  object  of  the 
resistance  leads  is  to  diminish  the  short-circuit  current  and  thus 
to  minimize  the  sparking  at  the  brushes,  the  losses  are  actually 
less  when  the  resistance  leads  are  used  than  when  they  are 
omitted.  This  fact  will  be  appreciated  when  it  is  remembered 
that  when  the  resistance  leads  are  not  used  the  loss  due  to  the 
short-circuit  current  is  enormous,  although  that  due  to  the  main 


SPARKING  IN  COMMUTATOR  MOTORS.  269 

line  current  may  be  small.  When  resistance  is  inserted  in  the 
coil  under  the  brush  the  former  loss  is  decreased  and  the  latter 
is  increased.  In  practice  the  inserted  resistance  is  given  a 
value  such  that  the  sum  of  the  two  losses  is  a  minimum,  which 
condition  exists  when  the  two  losses  are  equal. 

INTERNAL  RESISTANCE  LEADS. 

The  mechanical  arrangement  of  the  preventive  resistances 
have  not  caused  any  very  great  difficulty  in  the  construction 
of  motors.  Each  lead  is  so  placed  that  it  forms  a  non-inductive 
path  for  both  the  short-circuit  current  and  the  main  line  current, 
which  condition  is  conducive  to  sparkless  commutation.  Ac- 
cording to  one  method  of  construction,  special  slots  are  cut 
in  the  core  for  the  reception  of  the  leads.  According  to  another 
method,  the  leads  are  placed  in  the  same  slots  with  the  main 
armature  winding.  The  resistance  leads,  after  being  insulated, 
are  laid  in  the  bottom  of  the  slots,  one  terminal  of  each  lead 
passing  to  a  commutator  segment  and  the  other  to  a  tapping 
point  on  the  active  armature  winding  which  occupies  the  top 
portions  of  the  slots. 

Objections  which  have  been  urged  against  the  use  of  resist- 
ance leads  relate  to  the  power  absorbed  by  the  leads,  and  to 
the  fact  that,  as  ordinarily  arranged  on  the  armature,  the  re- 
sistance cannot  conveniently  be  varied  during  the  operation 
of  the  machine.  Furthermore,  although  the  leads  are  placed 
in  positions  where  it  is  difficult  to  repair  or  replace  them  in 
case  they  are  damaged,  practical  requirements  demand  that  the 
leads  be  of  limited  cross-section,  entailing  the  constant  danger 
that  they  will  burn  out.  Several  schemes  have  been  proposed 
for  overcoming  these  objections. 

EXTERNAL  RESISTANCE  LEADS  WITH  Two  COMMUTATORS. 

According  to  one  of  these  schemes  the  motor  is  provided 
with  two  commutators  connected  to  opposite  ends  of  the  arma- 
ture conductors,  each  commutator  having  alternate  live  and 
dead  segments.  The  brushes  bearing  on  each  commutator 
have  a  width  not  greater  than  that  of  a  commutator  segment. 
Several  brushes  of  each  polarity  are  distributed  around  each 
commutator,  the  brushes  being  so  arranged  that  when  one 
brush  is  on  a  dead  segment  other  brushes  of  the  same  polarity 


270  ALTERNATING  CURRENT  MOTORS. 

are  on  live  segments.  The  motor  is  arranged  to  be  started 
with  sufficient  resistance  between  the  parallel  connected  brushes 
to  limit  the  short-circuit  current  to  the  desired  amount,  and 
this  resistance  is  decreased  as  the  motor  comes  up  to  speed. 

EXTERNAL  RESISTANCE  LEADS  WITH  ONE  COMMUTATOR. 

Another  scheme  which  accomplishes  the  same  results  with 
the  use  of  only  one  commutator  is  indicated  diagrammatically 
in  Fig.  122.  Instead  of  the  usual  single  brush  or  set  of  brushes 
at  each  point  of  commutation  there  are  employed  three  inde- 
pendently insulated  brushes,  each  brush  having  a  width  some- 


FIG.  122. — External  Preventive  Resistance  for  Commutator  Motor. 

what  less  than  the  width  of  a  dead  segment.  The  outer  brushes 
are  connected  to  the  terminals  of  a  reactance  coil,  while  the 
middle  brush  is  connected  through  resistances  to  the  middle 
point  of  the  reactance  coil  and  to  a  terminal  of  the  machine. 
It  will  be  noted  that  when  the  outer  brushes  are  on  live  seg- 
ments, the  only  current  which  can  flow  in  the  local  circuit 
of  the  armature  coil  under  the  brushes  and  the  reactance  coil 
is  the  negligible  exciting  current  of  the  coil.  The  main  power 
current  flowing  through  the  armature  passes  differentially 
through  the  halves  of  the  reactance  coil  and  hence  causes  no 
opposing  reactance.  That  is  to  say,  for  the  line  current  the 
coil  acts  like  a  non-inductive  resistance,  but  for  the  local  short- 


SPARKING  IN  COMMUTATOR  MOTORS.  271 

circuit  current  it  acts  like  a  true  reactance  coil  to  decrease  the 
current  to  a  negligible  value. 

When  the  commutator  moves  to  a  position  where  the  middle 
brush  is  immediately  over  a  live  segment,  no  current  what- 
soever passes  through  the  reactance  coil,  while  the  middle 
brush  conveys  the  entire  line  current.  In  each  of  these  two 
positions  the  machine  is  devoid  of  any  short-circuiting  effect 
and  no  abnormal  heating  is  produced  at  any  point. 

When  the  commmutator  is  in  an  intermediate  position, 
however,  where  the  middle  and  one  outer  brush  are  simultane- 
ously on  the  same  live  segment  a  disadvantageous  short-circuit 
does  exist.  In  the  position  here  assumed  an  electromotive 
force  having  a  value  equal  to  one  half  of  that  of  one  armature 
coil  tends  to  circulate  a  current  locally  through  one-half  of 
the  reactance  coil  and  the  two  brushes  which  bear  on  a  single 
commutator  segment.  The  adjustable  resistances  inserted 
in  the  circuit  of  the  middle  brush  serve  to  keep  the  short-circuit 
current  within  proper  limits. 

It  will  be  noted  that  if  the  two  outer  brushes  be  placed  on 
a  single  commutator  segment,  the  reactance  coil  can  be  omitted 
and  yet  the  resistances  JR1  and  R2  may  be  employed  to  limit 
the  value  of  the  short-circuit  current.  In  this  latter  event 
there  are  in  effect  only  two  brushes  at  each  commutation  point 
and  there  occur  only  one-half  as  many  short-circuits  per  revo- 
lution as  occur  with  the  arrangement  shown  in  Fig.  122.  Sim- 
plicity would  seem  to  dictate  the  use  of  two  brushes  without 
the  reactance  coil  rather  than  three  brushes  with  the  coil.  A 
little  consideration  will  show  that  by  inserting  the  reactance 
coil  in  circuit  and  dividing  one  brush  into  two  parts  the  effect  as 
far  as  commutation  is  concerned  is  exactly  the  same  as  though 
each  segment  of  the  commutator  were  live  and  the  voltage  be- 
tween segments  were  reduced  to  one-half  of  the  value  actually 
produced  in  each  armature  coil ;  that  is,  the  reactance  coil  serves 
to  obtain  a  middle  e.m.f.  point  on  each  armature  coil,  and 
an  armature  provided  with  a  one-  turn-  per-  coil  winding 
commutates  as  though  there  were  only  one-half  turn  per  coil. 

In  the  arrangement  shown  in  Fig.  122  the  resistances  are  made 
of  ample  current  carrying  capacity  for  the  maximum  load  on 
the  machine,  and  they  are,  therefore,  not  subject  to  burn- 
outs. Moreover,  they  are  external  to  the  armature,  and  can 


.272  ALTERNATING  CURRENT  MOTORS. 

easily  be  adjusted  or  repaired.  The  reactance  coils  may  be 
located  at  any  convenient  distance  from  the  brushes  of  the 
machine  and  the  connecting  leads  will  serve  as  resistance  to 
limit  the  value  of  the  short-circuit  current.  Additional  re- 
sistance can  be  inserted  as  found  necessary,  this  latter  resistance 
being  varied  at  will  during  the  operation  of  the  machine.  Thus 
the  normal  "  starting  "  resistance  may  be  employed  simul- 
taneously to  limit  the  short-circuit  current  at  starting. 


Air  Gap,  effect  of  volume  on-excit- 
ing current  of  induction  mo- 
tors, 135. 

Alternators  (see  Generators). 

Angle  of  lag,  determination  of,  11. 
determination  of  with  one 
watt-meter,  6. 

Arc  lamps,  frequency  required  by, 
31. 

Armature  turns  effective,  216. 
windings  interlaced,  267. 

Auto  starter  for  induction  motors, 
22. 

Circle  diagram,  65,  100. 
accuracy  of,  105. 
errors  in,  71. 

Circuits,  electric  and  magnetic,  95. 
equivalent  electric,  98. 
magnetic    reluctance,     effect     of 

varying,  96. 

single-phase  and  polyphase,  1. 
Coils,  effect  of  grouping  on  capacity 

of  induction  motors,  133. 
in  series,  effective  value  of  e.m.f. 

in,  150. 

Commutating  motors  (see  Motors). 
Commutator    on    rotor    of    single- 
phase  induction  motor,  56. 
used  to  excite  asynchronous  gen- 
erators, 92. 

Concatenation  control,  23. 
Condensance,  adjustment  of,  89. 
used  as  source  of  exciting  cur- 
rent, 83. 
used  in  split-phase  motor,  62. 


Condensers,  operation  of,  84. 
Conducting  material,  economy  of,  1 
required   for   different    transmis- 
sion systems,   3. 

Control  of  induction  motor  by  con- 
catenation,  23. 
Converters,  frequency,  31. 
capacity  of,  33. 
field  of  application,  31. 
performance  of,  32. 
power  supplied  by,  33. 
inverted,  170. 
six-phase,  174. 
synchronous,  149. 

capacity,  relative  154. 

for  various  phases,  163. 
characteristics  of,  172. 
compounding  of,  169. 
current  (a.c.)  maximum,  value 

of,  155. 

equations  of,  173. 
excitation  of,  166. 
frequency  of,  31. 
heat  loss  in  armature  coils,  dis- 
tribution of,  159. 
hunting  of,  167. 
operation  at   fractional  power 

factor,    160. 

operation  at  unity  power  fac- 
tor, 154. 
performance,    characteristic, 

165. 

performance,    predetermina- 
tion of,  171. 
starting  of,    168. 
two-phase,  currents  in,  156. 


273 


274 


INDEX. 


Core,  effect  of  volume  on  exciting 
current  of  induction  motors, 
135. 

flux  in  induction  motors,  deter- 
mination of,  130. 

Currents,  equivalent  single-phase, 
13. 

Delta  vs.  star-connected  primaries, 

180. 

Diagram  of  compensated  repulsion 
motor,  221. 

of  compensated  series  motor,  with 
shunted  field  coil,  261. 

of  induction  series  motors,  246. 

of   inductively   compensated    se- 
ries motors,  240,  241. 

of  performance  of  polyphase  in- 
duction motors,    109. 

of    performance    of    single-phase 
induction  motors,  115. 

of  plain  series  motor,  233. 

of  repulsion  motor,  193-206. 
Dorble  current  generators  (see  gen- 
erators). 

Eddy  current  losses,  98. 
Effective,  armature  turns,  216. 
Electrical-space  degrees,  123. 
Electrical-time  degrees,  123. 
E.m.f.s,  generated  by  an  alternat- 
ing field,  199. 

in  group  of  coils  in  series,  effec- 
tive value  of,  150. 

in  six-phase  circuit,  2. 

in  three-phase  circuit.  2. 
Electromagnetic  torque,  181. 
Equivalent  circuits,  98. 

single-phase  currents,  13. 

single-phase  resistance,  14. 
Excitation  of  induction  motors,  35. 

of  synchronous  machines,  166. 
Exciting  watts  in  induction  motors, 
73. 

Ferraris'  method  of  treating  single- 
phase  induction  motors,  138. 

Fisher-Hinnen  device  for  starting 
induction  motors,  23. 


Four-phase,   156. 

Frequency  converters  (see  Convert- 
ers). 

effect  upon  alternators  in  paral- 
lel, 31. 

required  by  arc  lamps,  31. 
used  in  rotary  converters,  31. 

Generators,  asynchronous,   74. 

characteristic  performance,  79. 

core,  design  of,  91. 

commutator,  excitation  of,  92. 

compensation     for     inductive 
load,   94. 

connections  for  condenser  ex- 
citation,  89. 

current  diagram,  75. 

excitation,    characteristics    of, 
86. 

excitation  of,  81. 

excited  by  condensance,  83. 

lagging  current  load,  effect  of, 
90. 

leading  current  load,  90. 

load  characteristics  of,  90. 

operation  of,  75. 

operation     with     commutator 
excitation,   93. 

parallel  operation  of,  80. 

performance,  calculation  of,  77 

shunt  excited,  94. 
d.  c.,  capacity  compared  with  va- 
rious a.  c.  machines,  164. 
double  current,  149,  161. 

capacity  for  various  phases,  163. 

heat  loss  in  armature  coils,  dis- 
tribution of,  159. 
synchronous,  151. 

capacity  compared  with  d.  c. 
machines,    152. 

capacityforvarious  phases,  163. 

parallel  operation  of,  80. 

Heyland    asynchronous    generator, 

94. 

induction  motor  (see  Motors), 
motor,    winding,    double    three- 
phase  secondary,   46. 


INDEX. 


275 


Hunting,   167. 
Hysteresis  losses,  98. 

Induction  motor  (see  Motor,  Induc- 
tion) . 

Induction  series  motors  (see  Mo- 
tors, Series  Induction). 

Inverted  converters  (see  Converters) 

Leakage  reactance  in  compensated 

repulsion  motors,  229. 
in  induction  motors,  73. 
in  induction  series  motors,  252. 
in  repulsion  motors,  209. 
Loading  back  method  of  measuring 

torque,    185. 
Loads  balanced,  6. 
unbalanced,  9. 

Magnetic  distribution  in  polyphase 

motors,  125. 

Measurements  of  power,  3. 
Mechanical-space  degrees,  123. 
Methods,  graphical,  advantage  of, 

63. 

Motors,  commutator,  e.m.fs.,  gener- 
ated by  alternating  field,  199. 
power  factor,  260. 
power  lost  in  resistance  leads, 

268. 

sparking,  prevention  of,  266. 
torque,  production  of,  181. 
treatment  of,  simplified,  189. 
induction,  alternating   current 

in  secondary  coils,  39. 
capacity,  method  of  increasing, 

47. 

concatenation  control,  23. 
core   flux   as   affected   by   dis- 
tributed winding,  131. 
core  flux,  determination  of,  130 
current  diagram  at  all  speeds, 
75. 

locus,  65. 

locus,  effect  of  resistance  on, 
108. 

equation,  70. 
exact,  107. 


Motors,  induction,  continued. 

design,  effect  of  leakage  react- 
ance on,  73. 
direct    current    in    secondary 

coils,   38. 
efficiency,  21. 
equations,  analytic,   19. 
excitation  of,  35. 
exciting  watts,  value  of,  73. 
Fisher-Hinnen  device,  23. 
general  outline,  16. 
Heyland,  35. 

action    with    rotor    at    syn- 
chronous speed,  43. 
action  with  stationary  rotor, 

41. 

connecting  resistance,   func- 
tion of,  44. 
direct  current  armature  of, 

40. 

internal  voltage  diagram,  105. 
iron  losses,  98. 
magnetic  field  in,      121. 
method  of  treatment,  16. 
operation     above     synchron- 
ism,  75. 

below  synchronism,  74. 
outline  of  characteristic  feat- 
ures,  48. 
output,  21. 

performance  observed,  25. 
performance  diagram,  proof  of, 

111. 

polyphase,  capacity  of,  119. 
capacity  as  affected  by  grou- 
ping of  coils,  133. 
compared  with  single-phase 

motors,    112. 
exciting  current,  135. 
exciting  watts,  135. 
magnetic    distribution    with 

closed  secondary,   127. 
magnetic    distribution    with 

open  secondary,   124. 
magnetic  field  in,  121. 
output,  maximum,  110. 
operated  as  single-phase  ma- 
chines, 58. 


276 


INDEX. 


Motors,  induction,  polyphase,  con' 
tinned. 

performance,   complete   dia- 
gram  of,    109. 
power  factor,  maximum,  110 
torque,  maximum,  110. 
power  factor,  21. 
calculation  of,  29. 
maximum,  36. 
method  of  improving,  36. 
primary    current,     calculation 

of,   30. 

reactance  .at  standstill,  18. 
resistance  external  to  second- 
ary windings,  23. 
in  secondary  windings,  22. 
revolving  field,  production  of, 

17. 

secondary  current,  determina- 
tion of,  28, 

effective  resistance  in,  63. 
exciting  m.m.f.,  37. 
frequency,  18. 
resistance   measurement    of, 

28. 

series,  commutator,  245. 
single-phase,  48. 
capacity  of,  119. 
commutator  for  starting  pur- 
poses, 56. 
compared    with    polyphase, 

48,   112. 

currents  insecondary,49,143. 
electric  circuits  of,  114. 
equivalent    circuit     (exact), 

114. 
equivalent  circuit  (modified) 

115. 

magnetic  field  in,  138. 
magnetising  current,  54. 
performance,   diagram,    115. 
polyphase  motor,  used  as,  58 
quadrature  magnetism,  pro- 
duction of,  49. 
revolving  field,  elliptical,  53. 
circular,  52. 
production  of,  51. 
secondary  currents  in,  143. 


Motors,  induction,  single-phase, 
continued. 

secondary  quantities,  graph- 
ical representation  of,  146. 
shading  coils,  54. 
speed  equation,  117. 
speed  field,    117. 

as  affected  by  speed,  148. 
current  production  of,  139 
torque,  117. 

action   of  commutator  in 

producing,  56. 
starting,  53. 

transformer  features  of,  142. 
transformer  field,  as  affected 

by  speed,   148. 
slip  measurement  of,  26. 
starting  devices  for,  22. 
synchronous  speed,  determina- 
tion of,  24. 

method  of  decreasing,  24. 
tandem  control,  24. 
test   with   one   voltmeter   and 

one  wattmeter,  25. 
three-phase,   equivalent   start- 
ing current,   113. 
starting  current  (equivalent 

single-phase),  113. 
operated  on  single-phase  cir- 
cuit, 61. 
torque,  determination  of,  27. 

maximum,  20. 
transformer  features  of,  95. 
treatment  of,  graphical,  63. 
two-phase,  equivalent  starting 
current,  113. 

operated  on  single-phase  cir- 
cuit, 59. 
starting  current  (equivalent 

single-phase),  113. 
test  results,  (>6. 
used  as  frequency  converters, 
31. 

generators,  74. 
synchronous  motor,  39. 
voltage,  internal,  105. 
winding  distributed,    131. 
effect  of,  on  core-flux,  131. 


INDEX. 


277 


Motors,  continued. 
repulsion,  189. 

brush,    short-circuiting    effect, 

212. 

characteristics  of,  197-198. 
compensated,  214. 

brush,  short-circuiting  effect 
228. 

characteristics  of,  222. 

fundamental    equations    of, 
218. 

leakage  reactance,   227. 

performance,  calculation  of, 
223. 
observed,  224. 

resistance,  227. 

test,  225-226. 

vector  diagram  of,  221. 
construction  of,  201. 
diagram  proof  of,  208. 
electrical  circuits  (ideal),  190. 
equations  of,  210. 
fundamental  equations  of,  204. 
graphical  diagram  of,  193. 
graphically  treated,   199. 
impedance  apparent,  217. 
leakage  reactance,  209. 
magnetic  circuits  (ideal),   190. 
operation  of,  202. 
performance.calculation  of,  195. 

observed,  213. 
resistance  of,  209. 
test  of,  214. 
torque,  209. 

production  of,  192. 
treatment,  algebraic,  199. 
vector  diagram  of,  206. 
series,  232. 

compensated  conductively,  242 

equations,  243. 

impedance,  242. 

inductively,  241. 

line  current,  243. 

performance,  calculation  of, 
243. 

power,  243. 

power  factor,  242. 

torque,  243. 


Motors,  series,  compensated,  con- 
tinued. 

'   vector  diagram,  244. 
field  winding  shunted  with  re- 
sistance, 260. 
fundamental     equations     with 

non-uniform  reluctance,  237. 

with  uniform  air-gap  reluc- 
tance, 233. 

hysteresis  loss  in,  258. 
induction,  245. 

brush,  short-circuiting  effect 
255. 

equations,  fundamental,  247 

generator  action  in,  254. 

leakage  reactance,  252. 

resistance,  252. 

starting  torque,  250. 

test  of,  255. 

vector  diagram,  253. 
hysteretic  angle  of  time-phase 

displacemnet,  257. 
loss  in  shunted  resistance,  263. 
performance  with  shunted  field 

coils,  264. 
power,  236. 
power  factor,  235. 
resistance  in  shunt  with  field 

winding,   260. 
torque,  236. 
treatment,  algebraic,  232. 

graphical,  232. 
synchronous,  149,  151. 
capacities  relative,  154. 
excitation  of,  166. 
hunting  of,  167. 

Performance  diagram  for  polyphase 

induction  motors,  109. 
of  single-phase  induction  mo- 
tors, 115. 

Phase,  relation  of  voltages  in  syn- 
chronous machines,   150. 

Polyphase  circuits,  1. 

induction    motors    (see    Motors, 
Induction). 

Power,  apparent  in  three-phase  cir- 
cuit, 13. 


278 


INDEX. 


Power,  continued. 

factor,  adjusted  by  resistance  in 
shunt  with  field  winding,  260. 
determination  of,  11. 
maximum  of  induction  motors, 

36. 

method  of  improving,  36. 
measurements,  three-phase,  3. 
unbalanced     three-phase     cir- 
cuits,  3. 
three-phase,     one     wattmeter 

method,  6. 

two  wattmeter  method,  proof 
of,   3. 

Quadrature  watts,  8. 

Reactance,  leakage,  73. 

in  compensated  repulsion  mo- 
tors, 229. 

in  induction  series  motors,  252. 
in  repulsion  motors,  209. 
of     induction    motors     running, 

18. 

of    induction    motors   at    stand- 
still,   18. 
Resistance  in  secondary  winding  of 

induction  motor,  22. 
equivalent,  single-phase,  14. 
in  shunt  with  field  winding,  260. 
used  to  prevent  sparking,  267. 
Rotary  converters  (see  Converters) . 
Revolving  field,  production  of,  17. 

Shading  coils,  54. 
action  of,  54. 
Single-phase  circuits,  1. 

induction    motors    (see    Motors, 

Induction) . 

Six-phase  transformation,  175. 
Slip,  18. 

measurement  of,  26. 
Sparkingin  commutator  motors,  266. 
power  lost  in  resistance  leads,  268 
prevention  of,  by  external  resist- 
ance leads  with  one  commu- 
tator, 270. 

by    external    resistance    leads 
w^h  two  commutators,  269. 


Sparking,  prevention  of,  continued. 
by  internal  resistance  leads,  269- 
by  interlaced  windings,  267. 
by  series  resistance,  267. 
transformer  action  with  station- 
ary rotor,  266. 
Split  phase  motor,  58. 

use  of  condensance  with,  62. 
Star  vs.  delta-connected  primaries, 

180. 

Starting  devices  for  induction  mo- 
tors, 22. 

Steinmetz  method  of  treating  single 
phase  induction  motors,  139. 
Synchronous    commutating   ma- 
chines, definition  of,  149. 
converters   (see  Converters), 
motors  (see  Motors), 
speed,  determination  of,  24. 

Tandem  control  of  induction   mo- 
tors,  24. 

Three-phase  to  six-phase  transform- 
ation, 175. 

Torque,  determination  of,  27,  184. 
electromagnetic,  181. 
for  non-uniform  reluctance,  183. 
for  uniform  reluctance,  181. 
measurement  of,  185. 

electrical,  errors  in,  187. 
production    of    in    commutating 

motors,    181. 
Transformer  action  with  stationary 

rotor,  266. 

circle  diagram  for,  100. 
connections,  175. 
equivalent  circuits  of,  99. 
equivalent   circuits   of    (approxi- 
mate),  99. 

hysteresis  loss  in,  258. 
iron  losses,  98. 
principle  of,  95. 
six-phase,  175. 

used  to  adjust  condensance,  89. 
Two-phase  to  six-phase  transform- 
ation,   175. 

Windings,  armature,  interlaced, 267 
distributed,  131. 


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